3.19: Quarter-Wavelength Transmission Line
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Quarter-wavelength sections of transmission line play an important role in many systems at radio and optical frequencies. The remarkable properties of open- and short-circuited quarter-wave line are presented in Section 3.16 and should be reviewed before reading further. In this section, we perform a more general analysis, considering not just open- and short-circuit terminations but any terminating impedance, and then we address some applications.
The general expression for the input impedance of a lossless transmission line is (Section 3.15): \[Z_{in}(l) = Z_0 \frac{ 1 + \Gamma e^{-j2\beta l} }{ 1 - \Gamma e^{-j2\beta l} } \label{m0093_eZ} \] Note that when \(l=\lambda/4\): \[2\beta l = 2 \cdot \frac{2\pi}{\lambda} \cdot \frac{\lambda}{4} = \pi \nonumber \] Subsequently: \[\begin{split} Z_{in}(\lambda/4) &= Z_0 \frac{ 1 + \Gamma e^{-j\pi} }{ 1 - \Gamma e^{-j\pi} } \\ &= Z_0 \frac{ 1 - \Gamma }{ 1 + \Gamma } \end{split} \nonumber \] Recall that (Section 3.15): \[\Gamma = \frac{Z_L-Z_0}{Z_L+Z_0} \nonumber \] Substituting this expression and then multiplying numerator and denominator by \(Z_L+Z_0\), one obtains \[\begin{split} Z_{in}(\lambda/4) &= Z_0 \frac{ \left( Z_L + Z_0\right) - \left(Z_L - Z_0\right) }{ \left( Z_L + Z_0\right) + \left(Z_L - Z_0\right) } \\ &= Z_0 \frac{ 2Z_0 }{ 2Z_L } \end{split} \nonumber \] Thus, \[\boxed{ Z_{in}(\lambda/4) = \frac{Z_0^2 }{ Z_L } } \label{m0091_eQWII} \] Note that the input impedance is inversely proportional to the load impedance. For this reason, a transmission line of length \(\lambda/4\) is sometimes referred to as a quarter-wave inverter or simply as a impedance inverter .
Quarter-wave lines play a very important role in RF engineering. As impedance inverters, they have the useful attribute of transforming small impedances into large impedances, and vice-versa – we’ll come back to this idea later in this section. First, let’s consider how quarter-wave lines are used for impedance matching. Look what happens when we solve Equation \ref{m0091_eQWII} for \(Z_0\): \[Z_0 = \sqrt{Z_{in}(\lambda/4) \cdot Z_L } \label{m0091_eQWZ0} \] This equation indicates that we may match the load \(Z_L\) to a source impedance (represented by \(Z_{in}(\lambda/4)\)) simply by making the characteristic impedance equal to the value given by the above expression and setting the length to \(\lambda/4\). The scheme is shown in Figure \(\PageIndex{1}\).
Figure \(\PageIndex{1}\): Impedance-matching using a quarter-wavelength transmission line.Example \(\PageIndex{1}\): 300-to-\(50~\Omega\) match using an quarter-wave section of line
Design a transmission line segment that matches \(300~\Omega\) to \(50~\Omega\) at 10 GHz using a quarter-wave match. Assume microstrip line for which propagation occurs with wavelength 60% that of free space.
Solution
The line is completely specified given its characteristic impedance \(Z_0\) and length \(l\). The length should be one-quarter wavelength with respect to the signal propagating in the line. The free-space wavelength \(\lambda_0=c/f\) at 10 GHz is \(\cong 3\) cm. Therefore, the wavelength of the signal in the line is \(\lambda=0.6\lambda_0\cong 1.8\) cm, and the length of the line should be \(l=\lambda/4 \cong 4.5\) mm.
The characteristic impedance is given by Equation ref{m0091_eQWZ0}: \[Z_0 = \sqrt{ 300~\Omega \cdot 50~\Omega } \cong 122.5~\Omega \nonumber \] This value would be used to determine the width of the microstrip line, as discussed in Section 3.11.
It should be noted that for this scheme to yield a real-valued characteristic impedance, the product of the source and load impedances must be a real-valued number. In particular, this method is not suitable if \(Z_L\) has a significant imaginary-valued component and matching to a real-valued source impedance is desired. One possible workaround in this case is the two-stage strategy shown in Figure \(\PageIndex{2}\). In this scheme, the load impedance is first transformed to a real-valued impedance using a length \(l_1\) of transmission line. This is accomplished using Equation \ref{m0093_eZ} (quite simple using a numerical search) or using the Smith chart (see “Additional Reading” at the end of this section). The characteristic impedance \(Z_{01}\) of this transmission line is not critical and can be selected for convenience. Normally, the smallest value of \(l_1\) is desired. This value will always be less than \(\lambda/4\) since \(Z_{in}(l_1)\) is periodic in \(l_1\) with period \(\lambda/2\); i.e., there are two changes in the sign of the imaginary component of \(Z_{in}(l_1)\) as \(l_1\) is increased from zero to \(\lambda/2\). After eliminating the imaginary component of \(Z_L\) in this manner, the real component of the resulting impedance may then be transformed using the quarter-wave matching technique described earlier in this section.
Figure \(\PageIndex{2}\): Impedance-matching a complex-valued load impedance using quarter-wavelength transmission line.Example \(\PageIndex{2}\): Matching a patch Antenna to \(50~\Omega\)
A particular patch antenna exhibits a source impedance of \(Z_A = 35+j35~\Omega\). (See “Microstrip antenna” in “Additional Reading” at the end of this section for some optional reading on patch antennas.) Interface this antenna to \(50~\Omega\) using the technique described above. For the section of transmission line adjacent to the patch antenna, use characteristic impedance \(Z_{01}=50~\Omega\). Determine the lengths \(l_1\) and \(l_2\) of the two segments of transmission line, and the characteristic impedance \(Z_{02}\) of the second (quarter-wave) segment.
Solution
The length of the first section of the transmission line (adjacent to the antenna) is determined using Equation \ref{m0093_eZ}: \[Z_1(l_1) = Z_{01} \frac{ 1 + \Gamma e^{-j2\beta_1 l_1} }{ 1 - \Gamma e^{-j2\beta_1 l_1} } \nonumber \] where \(\beta_1\) is the phase propagation constant for this section of transmission line and \[\Gamma \triangleq \frac{Z_A-Z_{01}}{Z_A+Z_{01}} \cong -0.0059+j0.4142 \nonumber \] We seek the value of smallest positive value of \(\beta_1 l_1\) for which the imaginary part of \(Z_1(l_1)\) is zero. This can determined using a Smith chart (see “Additional Reading” at the end of this section) or simply by a few iterations of trial-and-error. Either way we find \(Z_1(\beta_1 l_1 = 0.793~\mbox{rad}) \cong 120.719-j0.111~\Omega\), which we deem to be close enough to be acceptable. Note that \(\beta_1 = 2\pi/\lambda\), where \(\lambda\) is the wavelength of the signal in the transmission line. Therefore \[l_1 = \frac{\beta_1 l_1}{\beta_1} = \frac{\beta_1 l_1}{2\pi} \lambda \cong 0.126\lambda \nonumber \]
The length of the second section of the transmission line, being a quarter-wavelength transformer, should be \(l_2 = 0.25\lambda\). Using Equation \ref{m0091_eQWZ0}, the characteristic impedance \(Z_{02}\) of this section of line should be \[Z_{02} \cong \sqrt{\left(120.719~\Omega\right) \left(50~\Omega\right) } \cong 77.7~\Omega \nonumber \]
Discussion. The total length of the matching structure is \(l_1+l_2 \cong 0.376\lambda\). A patch antenna would typically have sides of length about \(\lambda/2 = 0.5\lambda\), so the matching structure is nearly as big as the antenna itself. At frequencies where patch antennas are commonly used, and especially at frequencies in the UHF (300–3000 MHz) band, patch antennas are often comparable to the size of the system, so it is not attractive to have the matching structure also require a similar amount of space. Thus, we would be motivated to find a smaller matching structure.
Although quarter-wave matching techniques are generally effective and commonly used, they have one important contraindication, noted above – They often result in structures that are large. That is, any structure which employs a quarter-wave match will be at least \(\lambda/4\) long, and \(\lambda/4\) is typically large compared to the associated electronics. Other transmission line matching techniques – and in particular, single stub matching (Section 3.23) – typically result in structures which are significantly smaller.
The impedance inversion property of quarter-wavelength lines has applications beyond impedance matching. The following example demonstrates one such application:
Example \(\PageIndex{3}\): RF/DC decoupling in transistor amplifiers
Figure \(\PageIndex{3}\): Use an inductor to decouple the DC input power from the RF output signal at the output of a common-emitter RF amplifierTransistor amplifiers for RF applications often receive DC current at the same terminal which delivers the amplified RF signal, as shown in Figure \(\PageIndex{3}\). The power supply typically has a low output impedance. If the power supply is directly connected to the transistor, then the RF will flow predominantly in the direction of the power supply as opposed to following the desired path, which exhibits a higher impedance. This can be addressed using an inductor in series with the power supply output. This works because the inductor exhibits low impedance at DC and high impedance at RF. Unfortunately, discrete inductors are often not practical at high RF frequencies. This is because practical inductors also exhibit parallel capacitance, which tends to decrease impedance.
Figure \(\PageIndex{4}\): Decoupling of DC input power and RF output signal at the output of a common-emitter RF amplifier, using a quarter-wavelength transmission line.A solution is to replace the inductor with a transmission line having length \(\lambda/4\) as shown in Figure \(\PageIndex{4}\). A wavelength at DC is infinite, so the transmission line is essentially transparent to the power supply. At radio frequencies, the line transforms the low impedance of the power supply to an impedance that is very large relative to the impedance of the desired RF path. Furthermore, transmission lines on printed circuit boards are much cheaper than discrete inductors (and are always in stock!).
Additional Reading:
- “Quarter-wavelength impedance transformer” on Wikipedia.
- “Smith chart” on Wikipedia.
- “Microstrip antenna” on Wikipedia.


2.1.5: Mary Wollstonecraft (1759-1797)
A Vindication of the Rights of Woman British Romanticism More than any other woman in Britain, Mary Wollstonecraft represents the eighteenth-century movement in feminist thought. Wollstonecraft's alcoholic and abusive father failed to provide a consistent living for his family, so her childhood was spent in the turmoil of poverty and violence, and she had to earn a living early in life as a lady's companion, a teacher, a schoolmaster, a translator, and eventually a writer. Wollstonecraft's writing reflects her belief in the education of women as a remedy for inequality. She abhorred the conditions of women of all classes and the limited opportunities afforded them. Socially and politically active, Wollstonecraft became part of a group of radical dissenters who questioned the role of the individual in all phases of human life and espoused revolution as a means of liberty. She spent two years in France during the latter stage of the French Revolution, known as the Reign of Terror, and she recorded her observations in An Historical and Moral View of the Origins and Progress of the French Revolution; and the Effect it Has Produced in Europe (1794). In the treatise, Wollstonecraft expresses her disillusionment with a revolution which, in spite of assurances for total equality, excluded women from the political arena. Wollstonecraft faced a lifetime of personal and relational challenges; she survived as a single mother, only to die during the birth of her second daughter. That daughter, Mary Shelley, went on to create the classic Romantic novel Frankenstein . Foremost in Wollstonecraft's most famous and widely read work, A Vindication of the Rights of Woman (1792) is the call for equal rights for all human beings and equal education for men and women. Like her eighteenth-century predecessors, such as Mary Astell and Elizabeth Carter, Wollstonecraft admonishes her own sex against the frivolity that limits their ability to think rationally and weakens their characters. Her argument focuses on making women better wives and mothers through a combination of dignified treatment and intellectual encouragement. She also champions the right of women to participate in middle class work for financial security. The essay was well received in its initial publication, but its revolutionary ideas were not truly appreciated or realized until over one hundred years later, when the feminist movement revived Wollstonecraft's work and named her as the one of the most influential voices in the fight for women's rights. Consider while reading:
- Name some of the causes of what Wollstonecraft calls "women's degradation."
- What roles do women play in their own oppression?
Written by Karen Dodson


3.4: Expected Value and Law of Large Numbers
Would you buy a lottery ticket with the numbers 1, 2, 3, 4, 5? Do you think that a winning ticket with five consecutive numbers is less likely than a winning ticket with the numbers 2, 14, 18, 23 and 32? If you are playing a slot machine in Las Vegas and you have lost the last 10 times, do you keep playing the same machine because you are “due for a win?” Have you ever wondered how a casino can afford to offer meals and rooms at such cheap rates? Should you play a game of chance at a carnival? How much should an organization charge for raffle tickets for their next fund raiser? All of these questions can be answered using probabilities.
Suppose the random variable \(x\) can take on the \(n\) values \(x_{1}, x_{2}, x_{3}, \ldots, x_{n}\). If the probability that each of these values occurs is \(p_{1}, p_{2}, p_{3}, \ldots, p_{n}\), respectively, then the expected value of the random variable is
\[E(x)=x_{1} p_{1}+x_{2} p_{2}+x_{3} p_{3}+\ldots+x_{n} p_{n} \label{expectedvalue} \]
Valley View Elementary is trying to raise money to buy tablets for their classrooms. The PTA sells 2000 raffle tickets at $3 each. First prize is a flat-screen TV worth $500. Second prize is an android tablet worth $375. Third prize is an e-reader worth $200. Five $25 gift certificates will also be awarded. What are the expected winnings for a person who buys one ticket?
Solution
We need to write out the probability distribution before we find the expected value.
A total of eight tickets are winners and the other 1992 tickets are losers.
| Outcome | Win $500 | Win $375 | Win $200 | Win $25 | Win $0 |
|---|---|---|---|---|---|
| Probability | \(\dfrac{1}{2000}\) | \(\dfrac{1}{2000}\) | \(\dfrac{1}{2000}\) | \(\dfrac{5}{2000}=\dfrac{1}{400}\) | \(\dfrac{1992}{2000}=\dfrac{249}{250}\) |
Now use the formula for the expected value (Equation \ref{expectedvalue}).
\[\begin{aligned}
E &=\$500\left(\dfrac{1}{2000}\right)+ \$375\left(\dfrac{1}{2000}\right)+ \$200\left(\dfrac{1}{2000}\right)+ \$25\left(\dfrac{1}{400}\right)+ \$0\left(\dfrac{249}{250}\right) \\
&=0.60
\end{aligned} \nonumber \]
It costs $3 to buy a ticket but we only win an average of $0.60 per ticket. That means the expected winnings per ticket are $0.60 - $3 = -$2.40.
We would expect to lose an average of $2.40 for each ticket bought. This means that the school will earn an average of $2.40 for each ticket bought for a profit of $2.40 ∙ 2000 = $4800.
A real estate investor buys a parcel of land for $150,000. He estimates the probability that he can sell it for $200,000 to be 0.40, the probability that he can sell it for $160,000 to be 0.45 and the probability that he can sell it for $125,000 to be 0.15. What is the expected profit for this purchase?
Solution
Find the profit for each situation first. $200,000 – $150,000 = $50,000 profit, $160,000 - $150,000 = $10,000 profit, and $125,000 - $150,000 = -$25,000 profit (loss).
The probability distribution is
| Outcome | $50,000 | $10,000 | -$25,000 |
|---|---|---|---|
| Probability | 0.40 | 0.45 | 0.15 |
\[\begin{aligned}
E &=50,000(0.40)+10,000(0.45)+(-25,000)(0.15) \\
&=\$ 20,750
\end{aligned} \nonumber \]
The expected profit from the purchase is $20,750.
The cost of a $50,000 life insurance policy is $150 per year for a person who is 21-years old. Assume the probability that a person will die at age 21 is 0.001. What is the company’s expected profit if the company sells 10,000 policies to 21-year olds?
Solution
There are two outcomes. If the person lives the insurance company makes a profit of $150. The probability that the person lives is 1-0.001=0.999. If the person dies the company takes in $150 and pays out $50,000 for a loss of $49,850.
| Outcome | $150 | -$49,850 |
|---|---|---|
| Probability | 0.999 | 0.001 |
The expected value for one policy is:
\[E(x)=\operatorname{Sis} 0(0.999)+(-\$ 49,850)(0.001)=\$ 149.80 \nonumber \]
If the company sells 10,000 policies at a profit of $149.80 each, the total expected profit is .
A game that has an expected value of zero is called a fair game.
   Otherwise you lose $1.00. Is this a fair game? What would you expect would happen if you played the game many times?
Solution
First we need to find the probability distribution.
\[P(\text {both red})=\dfrac{5}{13} \cdot \dfrac{4}{12}=\dfrac{20}{156}=\dfrac{5}{39} \nonumber \]
\[P(\text {both white})=\dfrac{8}{13} \cdot \dfrac{7}{12}=\dfrac{56}{156}=\dfrac{14}{39} \nonumber \]
\[\begin{align*} P(\text{1 red and 1 white}) &= P(\text{red then white}) \, \text{or}\, (\text{white then red}) \\[4pt] &=P(\text{red then white}) + P(\text{white then red}) \\[4pt] &=\dfrac{5}{13} \cdot \dfrac{8}{12}+\dfrac{8}{13} \cdot \dfrac{5}{12} \\[4pt] &=\dfrac{80}{156}=\dfrac{20}{39} \end{align*} \nonumber \]
Check that the sum of the probabilities is 1.00.
\[\dfrac{5}{39}+\dfrac{14}{39}+\dfrac{20}{39}=\dfrac{39}{39}=1.00 \nonumber \]
Thus, the probability distribution is valid and is shown below:
| Outcome | Win $6.00 | Lose $1.50 | Lose $1.00 |
|---|---|---|---|
| Probability | \(\dfrac{5}{39}\) | \(\dfrac{14}{39}\) | \(\dfrac{20}{39}\) |
Now find the expected value (Equation \ref{expectedvalue}):
\[E=\dfrac{5}{39}(6.00)+\dfrac{14}{39}(-1.50)+\dfrac{20}{39}(-1.00) \approx-0.28 \nonumber \]
Since the expected value is not zero this is not a fair game.  We can predict what will happen if we play the game many times. We would expect to lose an average of $0.28 for every game we play. That means that the carnival will make an average of $0.28 for every game played. In this example we would refer to the carnival as the “house.”
Have you ever wondered how casinos make money when they advertise a 99% payback on their slot machines? The games in a casino are not fair games since the expected value is not zero. The expected value of the game for a gambler is a small negative number like -$0.01. For a particular game the gambler may win or the gambler may lose. It’s a random experiment and we cannot predict the outcome. What we can predict is what will happen if the gambler continues to play the game many times. If the expected value is -$0.01, the gambler will expect to lose an average of $0.01 for every game played. If he/she plays 100 games, he/she will expect to lose 100(0.01) = $1.00. Every penny the gambler loses the casino keeps. If hundreds of gamblers play hundreds of games each, every day of the year all those pennies add up to millions of dollars. The casino is referred to as the “house” and the $0.01 that the house expects to win for each game played is called the “house edge.”
The house edge is the amount that the house can expect to earn for each dollar bet.
A roulette wheel consists of 38 slots numbered 0, 00, and 1 through 36, evenly spaced around a wheel. The wheel is spun one direction and a ball is rolled around the wheel in the opposite direction. Eventually the ball will drop into one of the numbered slots. A player bets $1 on a single number. If the ball lands in the slot for that number, the player wins $35, otherwise the player loses the $1. Find the house edge for this type of bet.
Solution
The house edge is the expected value so we need to find the probability distribution and then the expected value.
There are 38 slots. One slot wins and the other 37 slots lose so
\[P(\text{win}) = \dfrac{1}{38}\) and \(P(\text{lose}) = \dfrac{37}{38}. \nonumber \]
The probability distribution is
| Outcome | Win $35 | Lose $1 |
|---|---|---|
| Probability | \(\dfrac{1}{38}\) | \(\frac{37}{38}\) |
The expected value is:
\[E = $35(\dfrac{1}{38}) + (-1)(\dfrac{37}{38}) \approx -$0.0526 \nonumber \]
The expected value of the game is -$0.0526. This means that the player would expect to lose an average of 5.26 cents for each game played. The house would win an average of 5.26 cents for each game played.
The house edge is 5.26 cents.
Gamblers’ Fallacy and Streaks
Often times a gambler on a losing streak will keep betting in the belief that his/her luck must soon change. Consider flipping a fair coin. Each toss of the coin is independent of all the other tosses. Assume the coin has landed heads up the last eight times. Some people erroneously believe that the coin is more likely to land tails up on the next toss. In reality, the coin still has a 50% chance of landing tails up. It does not matter what happened the last eight tosses.
The gambler’s fallacy is the mistaken belief that a streak of bad luck makes a person due for a streak of good luck.
Toss a fair coin seven times and record which side lands up. For example HHTHTTT would represent getting a head on the first, second, and fourth tosses and a tail on the other tosses. Is a streak of all heads less likely than the other possible outcomes?
Solution
As we will see in Section 3.5, there are 128 possible ways to toss a coin seven times. Some of the possibilities are \(HHTTHHT\), \(HTHTHTH\), \(HHHHTTT\), and \(HTTHTTH\). Because tossing coins are independent events and the coin is fair each of these 128 possibilities has the same probability.
\[P(\mathrm{HHTTHHT})=\dfrac{1}{128} \nonumber \]
\[P(\mathrm{HTHTHTH})=\dfrac{1}{128} \nonumber \]
etc.
This also means that the probability of getting all heads is
\[P(\mathrm{HHHHHHH})=\dfrac{1}{128}. \nonumber \]
Getting a streak of all heads has exactly the same probability as any other possible outcome.
Law of Large Numbers
When studying probabilities, many times the law of large numbers will apply. If you want to observe what the probability is of getting tails up when flipping a coin, you could do an experiment. Suppose you flip a coin 20 times and the coin comes up tails nine times. Then, using an empirical probability, the probability of getting tails is 9/20 = 45%. However, we know that the theoretical probability for getting tails should be 1/2 = 50%. Why is this different? It is because there is error inherent to sampling methods. However, if you flip the coin 100 times or 1000 times, and use the information to calculate an empirical probability for getting tails up, then the probabilities you will observe will become closer to the theoretical probability of 50%. This is the law of large numbers.
The law of large numbers means that with larger numbers of trials of an experiment the observed empirical probability of an event will approach the calculated theoretical probability of the same event.


9.5: Exercises
Exercise \(\PageIndex{1}\)
Is the concept of a contour integral well-defined if the integrand \(f(z)\) is non-differentiable along the contour? Why or why not?
Exercise \(\PageIndex{2}\)
In Section 9.4, we dealt with the integral \[\int_{-\infty}^\infty \frac{dx}{x^2 + 1}.\] Redo this calculation, but this time close the contour in the lower half-plane. Show that the result is the same.
- Answer
-
By analytic continuation, consider the integral \[I = \oint \frac{dz}{z^4 + 1},\] where the contour is closed in the upper half-plane (we could also choose to close below without changing the results). The contour integral over the large arc scales with the arc radius \(R\) as \(R^{-3}\), so it vanishes as \(R \rightarrow \infty\). Hence, \(I\) is exactly equal to the definite integral we are after.
To evaluate the loop integral, we need the poles of the integrand, which are the solutions to \(z^4 = -1\). Writing \(-1 = \exp(i\pi)\), we find that the roots are \(\exp(i\pi/4) \times \{\text{4-roots of unity}\}.\) These can be written in the Cartesian representation as \[\begin{align} z_1 &= \frac{1+i}{\sqrt{2}} \\ z_2 &= \frac{-1+i}{\sqrt{2}} \\ z_3 &= \frac{-1-i}{\sqrt{2}} \\ z_4 &= \frac{1-i}{\sqrt{2}}.\end{align}\] By closing the contour above, we enclose \(z_1\) and \(z_2\). Thus, by the residue theorem, \[\begin{align} I &= 2\pi i \left\{\left[\mathrm{Res} \left(\frac{1}{z^4 + 1}\right)\right]_{z=z_1} + \left[\mathrm{Res} \left(\frac{1}{z^4 + 1}\right)\right]_{z=z_2}\right\} \\ &= 2\pi i\left[\frac{1}{(z_1-z_2)(z_1-z_3)(z_1-z_4)} + \frac{1}{(z_2-z_1)(z_2-z_3)(z_2-z_4)}\right] \\ &= 2\pi i \left[\frac{\sqrt{8}}{(2)(2+2i)(2i)} + \frac{\sqrt{8}}{(-2)(2i)(-2+2i)}\right] \\ &= \frac{\sqrt{2}\pi i}{2} \left[\frac{1}{-1+i} + \frac{1}{1+i}\right] \\ &= \frac{\pi}{\sqrt{2}}.\end{align}\]
Exercise \(\PageIndex{3}\)
Calculate \[\int_{-\infty}^\infty dx\; \frac{1}{x^4 + 1}.\]
Exercise \(\PageIndex{4}\)
Calculate \[\int_{-\infty}^\infty dx\; \left[\frac{\sin(x)}{x}\right]^2.\]
Exercise \(\PageIndex{5}\)
Calculate \[\int_0^\infty dx \frac{x^{\lambda}}{x+1}, \;\;\mathrm{where}\; -1 < \lambda < 0.\] Hint: place the integrand’s branch cut along the positive real axis.
Exercise \(\PageIndex{6}\)
Solve the definite integral \[I = \int_0^{2\pi} \frac{d\phi}{\cos\phi+3},\] via the following steps. First, show that along a unit circle in the complex plane centered at the origin, \[\cos\phi = \frac{1}{2}\left(z+\frac{1}{z}\right),\] where \(z(\phi) = \exp(i\phi)\) . Then define a complex function \(f(z)\) such that the loop integral \(\oint f(z) \;dz,\) taken over the circular contour, is equal to \(I\) . Hence, calculate \(I\) .
- Answer
-
A unit circle centered at the origin can be parameterized by \(z = \exp(i\phi)\). Hence, along this circle, \[\begin{align} \cos\phi &= \frac{1}{2}\left(e^{i\phi} + e^{-i\phi}\right) \\ &= \frac{1}{2} \left(e^{i\phi} + \frac{1}{e^{i\phi}}\right) \\ &= \frac{1}{2}\left(z + \frac{1}{z}\right).\end{align}\] Also, \[\frac{dz}{d\phi} = iz.\] Let us denote this circular contour by \(\Gamma\). We want to find a function \(f(z)\) such that \[\oint_\Gamma f(z) \;dz = \int_0^{2\pi} \frac{d\phi}{\cos\phi+3}.\] The contour integral on the left side can be parameterized as \[\int_0^{2\pi} d\phi\; f\big(z(\phi)\big)\, \frac{dz}{d\phi}.\] Therefore, we want \(f(z)\) such that \[\begin{align} f\big(z(\phi)\big) \frac{dz}{d\phi} &= \frac{1}{\cos\phi+3} \\ = f(z) \;\,(i z) &= \frac{1}{\frac{1}{2}\left(z+\frac{1}{z}\right) + 3}.\end{align}\] After some algebra, we obtain \[f(z) = \frac{-2i}{z^2 + 6z + 1}.\] The denominator in \(f(z)\) has two roots, which are both real: \[\begin{align} z_+ &= -3 + 2\sqrt{2} = - 0.17157\dots \\ z_- &= -3 - 2\sqrt{2} = -5.8284\dots\end{align}\] Only the \(z_+\) pole is enclosed by the unit circle. Thus, we can use the residue theorem to evaluate the integral: \[\begin{align} I = \oint_\Gamma \frac{-2i}{z^2 + 6z + 1} \, dz &= 2\pi i \; \mathrm{Res}\left[\frac{-2i}{(z-z_+)(z-z_-)}\right]_{z = z_+} \\ &= 2\pi i \left(\frac{-2i}{z_+-z_-}\right) \\ &= \frac{4\pi}{\left(-3 + 2\sqrt{2}\right)-\left(-3 - 2\sqrt{2}\right)} \\ &= \frac{\pi}{\sqrt{2}}.\end{align}\]
Exercise \(\PageIndex{7}\)
Suppose \(f(z)\) is analytic everywhere in the upper half-plane, including the real line, and that its magnitude vanishes as \(1/|z|\) or faster as \(|z| \rightarrow \infty\) . Find the value of the principal-value integral \[\mathcal{P}\left[\int_{-\infty}^\infty \frac{f(x)}{x-a} dx \right],\] where \(a\) is some real constant. Hence, prove that the real and imaginary parts of \(f\) along the real line are related by \[\begin{align} \mathrm{Re}\big[f(x)\big] &= \;\;\,\frac{1}{\pi} \mathcal{P}\left[\int_{-\infty}^\infty \frac{\mathrm{Im}[f(w)]}{w-x} dw\right] \\ \mathrm{Im}\big[f(x)\big] &= -\frac{1}{\pi} \mathcal{P}\left[\int_{-\infty}^\infty \frac{\mathrm{Re}[f(w)]}{w-x} dw\right]. \end{align}\] These are called the Kramers-Kronig relations . In physics, these relations impose important constraints on the frequency dependence of the real and imaginary parts of the dielectric function (the square of the complex refractive index).
- Answer
-
To evaluate the principal-value integral \[I = \mathcal{P}\left[\int_{-\infty}^\infty \frac{f(x)}{x-a} dx \right],\] we introduce the following loop contour:
The solution procedure is very similar to the example worked out in Section 9.4. From the properties of \(f(z)\) given in the problem statement, we can conclude that (i) the integrand is analytic on and within the loop contour, so the residue theorem can be used; and (ii) the integrand vanishes quickly enough far from the origin so that, by Jordan’s lemma, the integral over \(\Gamma_2\) vanishes. Hence, \[I = i \pi f(a).\] By relabelling the dummy variables \(x \rightarrow y\) and \(a \rightarrow x\), we can write \[f(x) = -\frac{i}{\pi} \mathcal{P}\left[\int_{-\infty}^\infty \frac{f(y)}{y-x} dy \right].\] Let us now break up \(f\) into its real and imaginary parts: \[\begin{align} \mathrm{Re}[f(x)] + i \mathrm{Im}[f(x)] &= -\frac{i}{\pi} \mathcal{P}\left[\int_{-\infty}^\infty \frac{\mathrm{Re}[f(y)] + i \mathrm{Im}[f(y)]}{y-x} \;dy \right] \\ &= \frac{1}{\pi} \mathcal{P}\left[\int_{-\infty}^\infty \frac{\mathrm{Im}[f(y)] -i\mathrm{Re}[f(y)]}{y-x} \;dy \right].\end{align}\] Equating the real and imaginary parts of the two sides, we obtain the following two real equations, which are the Kramers-Kronig relations: \[\begin{align} \mathrm{Re}[f(x)] &= \;\;\,\frac{1}{\pi} \mathcal{P}\left[\int_{-\infty}^\infty \frac{\mathrm{Im}[f(y)]}{y-x} \;dy \right] \\ \mathrm{Im}[f(x)] &= -\frac{1}{\pi} \mathcal{P}\left[\int_{-\infty}^\infty \frac{\mathrm{Re}[f(y)]}{y-x} \; dy \right].\end{align}\]


11.9: Active vs. Passive Voice
Active and Passive Voice
Voice is a nebulous term in writing. It can refer to the general “feel” of the writing, or it can be used in a more technical sense. In this section, we will focus on the latter sense as we discuss active and passive voice.
You’ve probably heard of the passive voice—perhaps in a comment from an English teacher or in the grammar checker of a word processor. In both of these instances, you were (likely) guided away from the passive voice. Why is this the case? Why is the passive voice so hated? After all, it’s been used twice on this page already (three times now). When the passive voice is used to frequently, it can make your writing seem flat and drab. However, there are some instances where the passive voice is a better choice than the active.
So just what is the difference between these two voices? In the simplest terms, an active voice sentence is written in the form of “A does B.” (For example, “Carmen sings the song.”) A passive voice sentence is written in the form of “B is done by A.” (For example, “The song is sung by Carmen.”) Both constructions are grammatically sound and correct.
Let’s look at a couple more examples of the passive voice:
- I’ve been hit! ( or , I have been hit!)
- Jasper was thrown from the car when it was struck from behind.
You may have noticed something unique about the previous two sentences: the subject of the sentence is not the person (or thing) performing the action. The passive voice “hides” who does the action. Despite these sentences being completely grammatically sound, we don’t know who hit “me” or what struck the car.
The passive is created using the verb to be (e.g., the song is sung; it was struck from behind). Remember that to be conjugates irregularly. Its forms include am, are, is, was, were, and will be , which we learned about earlier in the course.
Remember, to be also has more complex forms like had been , is being , and was being.
- Mirella is being pulled away from everything she loves.
- Pietro had been pushed; I knew it.
- Unfortunately, my car was being towed away by the time I got to it.
Because to be has other uses than just creating the passive voice, we need to be careful when we identify passive sentences. It’s easy to mistake a sentence like “She was falling.” or “He is short.” for a passive sentence. However, in “She was falling,” was simply indicates that the sentence takes place in the past. In “He is short,” is is a linking verb. If there is no “real” action taking place, is is simply acting as a linking verb.
There are two key features that will help you identify a passive sentence:
- Something is happening (the sentence has a verb that is not a linking verb).
- The subject of the sentence is not doing that thing.
Active vs Passive Voice in Your Writing. Provided by: GCFLearnFree.org. License: All Rights Reserved. License Terms: Standard YouTube License.
Usage
As you read at the two sentences below, think about the how the different voice may affect the meaning or implications of the sentence:
- Passive voice: The rate of evaporation is controlled by the size of an opening.
- Active voice: The size of an opening controls the rate of evaporation.
The passive choice slightly emphasizes “the rate of evaporation,” while the active choice emphasizes “the size of an opening.” Simple. So why all the fuss? Because passive constructions can produce grammatically tangled sentences such as this:
Groundwater flow is influenced by zones of fracture concentration, as can be recognized by the two model simulations (see Figures 1 and 2), by which one can see . . .
The sentence is becoming a burden for the reader, and probably for the writer too. As often happens, the passive voice here has smothered potential verbs and kicked off a runaway train of prepositions. But the reader’s task gets much easier in the revised version below:
Two model simulations (Figures 1 and 2) illustrate how zones of fracture concentration influence groundwater flow. These simulations show . . .
To revise the above, all I did was look for the two buried things (simulations and zones) in the original version that could actually do something, and I made the sentence clearly about these two nouns by placing them in front of active verbs. This is the general principle to follow as you compose in the active voice: Place concrete nouns that can perform work in front of active verbs.
Exercise 1
Are the following sentences in the active or passive voice?
- Jayden drank more sodas than anyone else at the party.
- The samples were prepared in a clean room before being sent out for further examination.
- Karen was dancing with Joshua when she suddenly realized she needed to leave.
- Carlos was a very serious scientist with unique interests.
- When I returned to my room, my luggage had been stolen.
Show Answer
- Answer
- This sentence uses the active voice. Jayden does the action ( drank ) to the object ( more sodas ). If this sentence were written in the passive it would read “More sodas were drunk by Jayden than by anyone else at the party.” This sentence uses the passive voice. The action ( prepared ) was done to the subject of the sentence ( samples ). If this sentence were written in the active it would be something like this: “[Actor] prepared the samples in a clean room before sending them out for further examination.” Since we do not know who prepared the samples, the active sentence is incomplete. This sentence uses the active voice. In this case was indicates that the sentence happened in the past; it does not indicate the passive voice in this instance. This sentence uses the active voice. In this case was is acting as a linking verb. It links Carlos with the phrase very serious scientist . The introductory phrase to the sentence (When I returned to my room) is in the active voice. The second phrase (my luggage had been stolen) uses the passive voice.
Revising Weak Passive-Voice Sentences
As we’ve mentioned, the passive voice can be a shifty operator—it can cover up its source, that is, who’s doing the acting, as this example shows:
-
Passive:
The papers
will be graded
according to the criteria stated in the syllabus.
- Graded by whom though?
- Active: The teacher will grade the papers according to the criteria stated in the syllabus.
It’s this ability to cover the actor or agent of the sentence that makes the passive voice a favorite of people in authority—policemen, city officials, and, yes, teachers. At any rate, you can see how the passive voice can cause wordiness, indirectness, and comprehension problems.
| Passive | Question | Active |
|---|---|---|
| Your figures have been reanalyzed in order to determine the coefficient of error. The results will be announced when the situation is judged appropriate. | Who analyzes, and who will announce? | We have reanalyzed your figures in order to determine the range of error. We will announce the results when the time is right. |
| With the price of housing at such inflated levels, those loans cannot be paid off in any shorter period of time. | Who can’t pay the loans off? | With the price of housing at such inflated levels, homeowners cannot pay off those loans in any shorter period of time. |
| After the arm of the hand-held stapler is pushed down, the blade from the magazine is raised by the top-leaf spring, and the magazine and base. | Who pushes it down, and who or what raises it? | After you push down on the arm of the hand-held stapler, the top-leaf spring raises the blade from the magazine, and the magazine and base move apart. |
| However, market share is being lost by 5.25-inch diskettes as is shown in the graph in Figure 2. | Who or what is losing market share, who or what shows it? | However, 5.25-inch diskettes are losing market share as the graph in Figure 2 shows. |
| For many years, federal regulations concerning the use of wire-tapping have been ignored . Only recently have tighter restrictions been imposed on the circumstances that warrant it. | Who has ignored the regulations, and who is now imposing them? | For many years, government officials have ignored federal regulations concerning the use of wire-tapping. Only recently has the federal government imposed tighter restrictions on the circumstances that warrant it. |
Exercise 2
Convert these passive voice sentences into the active voice. Why is the active voice a better choice for each of these sentences?
- The process, which was essential for the experiment’s success, was completed by Enzo.
- The cake that I worked on all day long is being eaten by Justin.
- After the pattern has been applied to the fabric, work on the embroidery can be started.
- Answer
-
Enzo completed the process, which was essential for the experiment’s success.
- In the passive sentence, the which -clause makes the subject of the sentence excessively long. By converting the sentence to the active voice, the clause is moved to the predicate, which makes the sentence easier to understand.
- The active voice works better in this sentence for the same reasons as sentence one. It is also likely that you would want to put emphasis on Justin in this sentence. After all, he’s doing something that is (most likely) inconsiderate.
- This sentence is likely found in a set of instructions, which are usually written directly to the reader. Addressing “you” and avoiding the passive voice will make the instructions feel more natural and accessible.
Don’t get the idea that the passive voice is always wrong and should never be used. It is a good writing technique when we don’t want to be bothered with an obvious or too-often-repeated subject and when we need to rearrange words in a sentence for emphasis. The next page will focus more on how and why to use the passive voice.
Using the Passive Voice
There are several different situations where the passive voice is more useful than the active voice.
-
When you don’t know who did the action:
The paper had been moved.
- The active voice would be something like this: “Someone had moved the paper.” While this sentence is technically fine, the passive voice sentence has a more subtle element of mystery, which can be especially helpful in creating a mood in fiction.
-
When you want to hide who did the action:
The window had been broken.
- The sentence is either hiding who broke the window or they do not know. Again, the sentence can be reformed to say “Someone had broken the window,” but using the word someone clearly indicates that someone (though we may not know who) is at fault here. Using the passive puts the focus on the window rather than on the person who broke it, as he or she is completely left out of the sentence.
-
When you want to emphasize the person or thing the action was done to:
Caroline was hurt when Kent broke up with her.
- We automatically focus on the subject of the sentence. If the sentence were to say “Kent hurt Caroline when he broke up with her,” then our focus would be drawn to Kent rather than Caroline.
-
A subject that can’t actually
do
anything:
Caroline was hurt when she fell into the trees.
- While the trees hurt Caroline, they didn’t actually do anything. Thus, it makes more sense to have Caroline as the subject rather than saying “The trees hurt Caroline when she fell into them.”
Note: It’s often against convention in scholarly writing to use I . While this may seem like a forced rule, it also stems from the fact that scholars want to emphasize the science or research as opposed to the author of the paper. This often results in the passive voice being the best choice. This is not the case in other formal settings, such as in resumes and in cover letters.
Exercise 3
Consider the following instances. In each case, determine why the writers might want to use active or passive voice. Write an example sentence based on their circumstances.
- Antonella made an error in her calculations that ruined an experiment. This error ended up costing both time and materials. She has to write a report to her boss. What might she say about the experiment?
- Isabel is writing a supernatural thriller. Her main character, Liam, notices that his keys aren’t where he left them. How might Isabel word this realization?
- Thiago is writing a cover letter to apply for a new job. He is listing out tasks that he does at his current job. How would he want to word these items?
- Answer
-
Antonella would likely want to write in the passive voice. Even if her boss knows she made the error, writing in the passive will draw attention away from that fact. She might say something like this:
- An error was made that ended up costing time and resources. The experiment will have to be repeated with new materials.
- Liam’s keys had been moved when he wasn’t looking.
- Something—or someone—had moved Liam’s keys when he wasn’t looking.
- I currently work as a teaching assistant for a linguistics professor. I organize her mail, flagging important items so she knows what needs immediate attention; I aid her in her research, finding interesting articles and studies; and I often help her students when her attention is needed elsewhere.
Using the Passive
Now that we know there are some instances where passive voice is the best choice, how do we use the passive voice to it fullest? The answer lies in writing direct sentences—in passive voice—that have simple subjects and verbs. Compare the two sentences below:
- Photomicrographs were taken to facilitate easy comparison of the samples.
- Easy comparison of the samples was facilitated by the taking of photomicrographs.
Both sentences are written in the passive voice, but for most ears the first sentence is more direct and understandable, and therefore preferable. Depending on the context, it does a clearer job of telling us what was done and why it was done. Especially if this sentence appears in the “Experimental” section of a report (and thus readers already know that the authors of the report took the photomicrographs), the first sentence neatly represents what the authors actually did—took photomicrographs—and why they did it—to facilitate easy comparison.
Exercise 4
Read the following sentences. Are they using the passive effectively? If there are any errors, rewrite the sentences accordingly.
- The machine needs to be reset at 10:23, 11:12, and 11:56 every night.
- The final steps, which need to be finished before the sun sets over the mountains, are going to be completed by Kajuana.
- The difficult task of measuring minute fluctuations in weight was made easier by the use of a new digital scale.
- Answer
-
Yes. In this case, it doesn’t matter who accomplishes the action; it simply needs to be done. If this sentence appears in an academic article, the passive may be even more appropriate, as that style often demands the actor be left out of the sentence. No. This would be better in the active voice. There are a lot of different parts to the sentence, and by converting the sentence to the active voice, they come in a more logical order that is easier to understand:
- Kajuana is going to complete the final steps, which need to be finished before the sun sets over the mountains.
- A new digital scale made it easier to measure minute fluctuations in weight.
As we mentioned in Participles , the passive voice can also be used following relative pronouns like that and which .
- I moved into the house that was built for me.
- Adrián’s dog loves the treats that are given to him.
- Brihanna has an album that was signed by the Beastie Boys.
In each of these sentences, it is grammatically sound to omit (or elide ) the pronoun and to be . Elision is used with a lot of different constructions in English; we use it shorten sentences when things are understood. However, we can only use elision in certain situations, so be careful when removing words! You may find these elided sentences more natural:
- I moved into the house built for me.
- Adrián’s dog loves the treats given to him.
- Brihanna has an album signed by the Beastie Boys
Additional Practice Exercises
Exercise 5
Convert these passive voice sentences into the active voice:
- Alana’s toes were crushed by the garage door.
- The passive voice has likely been heard of by you.
- Rebeca’s favorite spot in the lecture hall had been taken by the time she got to class.
- When the passive voice is overused, you often end up with flat writing.
Answer
The garage door crushed Alana’s toes. You’ve likely heard of the passive voice. Because there’s a descriptive phrase, there are a few options when revising this sentence:
- Someone had taken Rebeca’s favorite spot in the lecture hall by the time she got to class.
- By the time Rebeca got to class, someone had taken her favorite spot in the lecture hall.
Exercise 6
Read the following sentences. Are they using the passive effectively? Or should they be rewritten as active sentences?
- Maren was hit by several branches as she slid down the hill.
- A lot of discussion about whether technology is hurting or helping our ability to communicate has been inspired by this increase in technology.
- Listeners are encouraged by the lyrics to cast aside their fear and be themselves.
Answer
Yes, this sentence uses the passive effectively. Since the subject of this sentence— several branches —can’t actually do anything, it’s best to put the emphasis on Maren , the person the actions were done to. This sentence does not use the passive well. The passive voice has made the sentence a lot more complicated then it needs to be. The active would read something like this:
- This increase of technology has inspired a lot of discussion about whether technology is hurting or helping our ability to communicate.
- Passive: Listeners are encouraged by the lyrics to cast aside their fear and be themselves.
- Active: The lyrics encourage listeners to cast aside their fear and be themselves
Contributors and Attributions
- Adapted from Using the Passive Voice in Guide to Writing. Provided by: Lumen Learning. License: CC-BY-NC-SA.
This page most recently updated on June 8, 2020.


10.2: Magnets
Learning Objectives
- Describe the difference between the north and south poles of a magnet.
- Describe how magnetic poles interact with each other.
All magnets attract iron, such as that in a refrigerator door. However, magnets may attract or repel other magnets. Experimentation shows that all magnets have two poles. If freely suspended, one pole will point toward the north. The two poles are thus named the north magnetic pole and the south magnetic pole (or more properly, north-seeking and south-seeking poles, for the attractions in those directions).
UNIVERSAL CHARACTERISTICS OF MAGNETS AND MAGNETIC POLES
It is a universal characteristic of all magnets that like poles repel and unlike poles attract . (Note the similarity with electrostatics: unlike charges attract and like charges repel.)
Further experimentation shows that it is impossible to separate north and south poles in the manner that + and − charges can be separated.
MISCONCEPTION ALERT: EARTH’S GEOGRAPHIC NORTH POLE HIDES AN S
The Earth acts like a very large bar magnet with its south-seeking pole near the geographic North Pole. That is why the north pole of your compass is attracted toward the geographic north pole of the Earth—because the magnetic pole that is near the geographic North Pole is actually a south magnetic pole! Confusion arises because the geographic term “North Pole” has come to be used (incorrectly) for the magnetic pole that is near the North Pole. Thus, “North magnetic pole” is actually a misnomer—it should be called the South magnetic pole.
The fact that magnetic poles always occur in pairs of north and south is true from the very large scale—for example, sunspots always occur in pairs that are north and south magnetic poles—all the way down to the very small scale. Magnetic atoms have both a north pole and a south pole, as do many types of subatomic particles, such as electrons, protons, and neutrons.
MAKING CONNECTIONS: TAKE-HOME EXPERIMENT—REFRIGERATOR MAGNETS
We know that like magnetic poles repel and unlike poles attract. See if you can show this for two refrigerator magnets. Will the magnets stick if you turn them over? Why do they stick to the door anyway? What can you say about the magnetic properties of the door next to the magnet? Do refrigerator magnets stick to metal or plastic spoons? Do they stick to all types of metal?
Section Summary
- Magnetism is a subject that includes the properties of magnets, the effect of the magnetic force on moving charges and currents, and the creation of magnetic fields by currents.
- There are two types of magnetic poles, called the north magnetic pole and south magnetic pole.
- North magnetic poles are those that are attracted toward the Earth’s geographic north pole.
- Like poles repel and unlike poles attract.
- Magnetic poles always occur in pairs of north and south—it is not possible to isolate north and south poles.
Glossary
- north magnetic pole
- the end or the side of a magnet that is attracted toward Earth’s geographic north pole
- south magnetic pole
- the end or the side of a magnet that is attracted toward Earth’s geographic south pole


12.1: Accessing Individual Elements
Retrieving an Element
To get the value of a specific element from an array, we use “boxie notation” with the index number:
Code \(\PageIndex{1}\) (Python):
print(prez_elections[0])
third_year = usa_years[2]
print("{} was the 3rd year of U.S.A.".format(third_year))
print("The highest-numbered player is {}".format( roster[len(roster)-1]))
| 788.0
| 1778 was the 3rd year of U.S.A.
| The highest-numbered player is Christen Press.
Remember, indices start at zero (not one) so that’s why the first line has a 0 in it.
Now examine that last line, which is kind of tricky. Whenever you have boxies, you have to first evaluate the code inside them to get a number. That number is then the index number Python will look up in the array. In the last line above, the code inside the boxies is:
...len(roster)-1...
Breaking it down, we know that len(roster) is 24, which means len(roster)-1 must be 23, and so roster[len(roster)-1] is Christen Press . It’s a common pattern to get the last element of an array. 1
To test your understanding, figure out what the following code will print:
Code \(\PageIndex{2}\) (Python):
q = 2
r = np.array([45,17,39,99])
s = 1
print(r[q-s+1]+3)
The answer is at the end of the chapter.
Changing an Element
To modify an element of an array, just use the equals sign like we do for variables:
Code \(\PageIndex{3}\) (Python):
stooges = np.array(['Larry','Beavis','Moe'])
print(stooges)
stooges[1] = 'Curly'
print(stooges)
| ['Larry' 'Beavis' 'Moe']
| ['Larry' 'Curly' 'Moe']
After all, an individual element like stooges[1] is itself a variable pretty much like any other.
Slices
Sometimes, instead of getting just one element from an array, we want to get a whole chunk of them at a time. We’re interested in the first ten elements, say, or the last twenty, or the fourteen elements starting at index 100. To do this sort of thing, we use a slice .
Suppose we had a list of states in alphabetical order, and wanted to snag a chunk of consecutive entries out of the middle – say, Arizona through Colorado. Consider this code:
Code \(\PageIndex{4}\) (Python):
states = np.array(["AL","AK","AZ","AR","CA","CO","CT", "DE","FL","GA","HI"])
print(states[2:6])
| ['AZ' 'AR' 'CA' 'CO']
The “ 2:6 ” in the boxies tells Python that we want a slice with elements 2 through 5 (not through 6, as you’d expect). This is the same behavior we saw for np.arange() where the range goes up to, but not including, the last value. Just get used to it.
We can also omit the number before the colon, which tells Python to start at the beginning of the array:
Code \(\PageIndex{5}\) (Python):
print(states[:5])
| ['AL' 'AK' 'AZ' 'AR' 'CA']
or omit the number after the colon, which says to go until the end:
Code \(\PageIndex{6}\) (Python):
print(states[8:])
| ['FL' 'GA' 'HI']
We can even include a second colon, after which a third number specifies a step size, or stride length. Consider:
Code \(\PageIndex{7}\) (Python):
print(states[2:9:3])
| ['AZ' 'CO' 'FL']
This tells Python: “start the slice at element #2, and go up to (but not including) element #9, by threes .” If you count out the states by hand, you’ll see that Arizona is at index 2, Colorado is at index 5, and Florida is at index 8. Hence these are the three elements included in the slice.
This slice stuff may seem esoteric, but it comes up surprisingly often.


7.5: The Electromagnetic Spectrum
-
- Last updated
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Learning Objectives
By the end of this section, you will be able to:
- Explain how electromagnetic waves are divided into different ranges, depending on wavelength and corresponding frequency
- Describe how electromagnetic waves in different categories are produced
- Describe some of the many practical everyday applications of electromagnetic waves
Electromagnetic waves have a vast range of practical everyday applications that includes such diverse uses as communication by cell phone and radio broadcasting, WiFi, cooking, vision, medical imaging, and treating cancer. In this module, we discuss how electromagnetic waves are classified into categories such as radio, infrared, ultraviolet, and so on. We also summarize some of the main applications for each range.
The different categories of electromagnetic waves differ in their wavelength range, or equivalently, in their corresponding frequency ranges. Their properties change smoothly from one frequency range to the next, with different applications in each range. A brief overview of the production and utilization of electromagnetic waves is found in Table \(\PageIndex{1}\).
| Type of wave | Production | Applications | Issues |
|---|---|---|---|
| Radio | Accelerating charges | Communications, Remote controls, MRI | Requires control for band use |
| Microwaves | Accelerating charges and thermal agitation | Communications, Ovens, Radar, Cell phone use | |
| Infrared | Thermal agitation and electronic transitions | Thermal imaging, Heating | Absorbed by atmosphere, Greenhouse effect |
| Visible light | Thermal agitation and electronic transitions | Photosynthesis, Human vision | |
| Ultraviolet | Thermal agitation and electronic transitions | Sterilization, Vitamin D production | Ozone depletion, Cancer causing |
| X-rays | Inner electronic transitions and fast collisions | Security, Medical diagnosis, Cancer therapy | Cancer causing |
| Gamma rays | Nuclear decay | Nuclear medicine, Security, Medical diagnosis, Cancer therapy | Cancer causing, Radiation damage |
The relationship \(c = f\lambda\) between frequency f and wavelength \(\lambda\) applies to all waves and ensures that greater frequency means smaller wavelength. Figure \(\PageIndex{2}\) shows how the various types of electromagnetic waves are categorized according to their wavelengths and frequencies - that is, it shows the electromagnetic spectrum.
Radio Waves
The term radio waves refers to electromagnetic radiation with wavelengths greater than about 0.1 m. Radio waves are commonly used for audio communications (i.e., for radios), but the term is used for electromagnetic waves in this range regardless of their application. Radio waves typically result from an alternating current in the wires of a broadcast antenna. They cover a very broad wavelength range and are divided into many subranges, including microwaves, electromagnetic waves used for AM and FM radio, cellular telephones, and TV signals.
There is no lowest frequency of radio waves, but ELF waves, or “extremely low frequency” are among the lowest frequencies commonly encountered, from 3 Hz to 3 kHz. The accelerating charge in the ac currents of electrical power lines produce electromagnetic waves in this range. ELF waves are able to penetrate sea water, which strongly absorbs electromagnetic waves of higher frequency, and therefore are useful for submarine communications.
In order to use an electromagnetic wave to transmit information, the amplitude, frequency, or phase of the wave is modulated , or varied in a controlled way that encodes the intended information into the wave. In AM radio transmission, the amplitude of the wave is modulated to mimic the vibrations of the sound being conveyed. Fourier’s theorem implies that the modulated AM wave amounts to a superposition of waves covering some narrow frequency range. Each AM station is assigned a specific carrier frequency that, by international agreement, is allowed to vary by \(\pm 5 \, kHz\). In FM radio transmission, the frequency of the wave is modulated to carry this information, as illustrated in Figure \(\PageIndex{2}\), and the frequency of each station is allowed to use 100 kHz on each side of its carrier frequency. The electromagnetic wave produces a current in a receiving antenna, and the radio or television processes the signal to produce the sound and any image. The higher the frequency of the radio wave used to carry the data, the greater the detailed variation of the wave that can be carried by modulating it over each time unit, and the more data that can be transmitted per unit of time. The assigned frequencies for AM broadcasting are 540 to 1600 kHz, and for FM are 88 MHz to108 MHz.
Cell phone conversations, and television voice and video images are commonly transmitted as digital data, by converting the signal into a sequence of binary ones and zeros. This allows clearer data transmission when the signal is weak, and allows using computer algorithms to compress the digital data to transmit more data in each frequency range. Computer data as well is transmitted as a sequence of binary ones and zeros, each one or zero constituting one bit of data.
Microwaves
Microwaves are the highest-frequency electromagnetic waves that can be produced by currents in macroscopic circuits and devices. Microwave frequencies range from about \(10^{9}Hz\) to nearly \(10^{12} Hz\). Their high frequencies correspond to short wavelengths compared with other radio waves—hence the name “microwave.” Microwaves also occur naturally as the cosmic background radiation left over from the origin of the universe. Along with other ranges of electromagnetic waves, they are part of the radiation that any object above absolute zero emits and absorbs because of thermal agitation , that is, from the thermal motion of its atoms and molecules.
Most satellite-transmitted information is carried on microwaves . Radar is a common application of microwaves. By detecting and timing microwave echoes, radar systems can determine the distance to objects as diverse as clouds, aircraft, or even the surface of Venus.
Microwaves of 2.45 GHz are commonly used in microwave ovens. The electrons in a water molecule tend to remain closer to the oxygen nucleus than the hydrogen nuclei (Figure \(\PageIndex{3}\)). This creates two separated centers of equal and opposite charges, giving the molecule a dipole moment . The oscillating electric field of the microwaves inside the oven exerts a torque that tends to align each molecule first in one direction and then in the other, with the motion of each molecule coupled to others around it. This pumps energy into the continual thermal motion of the water to heat the food. The plate under the food contains no water, and remains relatively unheated.
The microwaves in a microwave oven reflect off the walls of the oven, so that the superposition of waves produces standing waves, similar to the standing waves of a vibrating guitar or violin string ( Normal Modes of a Standing Sound Wave ). A rotating fan acts as a stirrer by reflecting the microwaves in different directions, and food turntables, help spread out the hot spots.
Example \(\PageIndex{1}\): Why Microwave Ovens Heat Unevenly
How far apart are the hotspots in a 2.45-GHz microwave oven?
Strategy
Consider the waves along one direction in the oven, being reflected at the opposite wall from where they are generated.
Solution
The antinodes, where maximum intensity occurs, are half the wavelength apart, with separation
\[ \begin{align} d &= \dfrac{1}{2} \lambda \\[4pt] &= \dfrac{1}{2} \dfrac{c}{f} \\[4pt] &= \dfrac{3.00 \times 10^8 m/s}{2(2.45 \times 10^9 \, Hz)} \\[4pt] &= 6.02 \, cm. \end{align} \nonumber \]
Significance
The distance between the hot spots in a microwave oven are determined by the wavelength of the microwaves.
A cell phone has a radio receiver and a weak radio transmitter, both of which can quickly tune to hundreds of specifically assigned microwave frequencies. The low intensity of the transmitted signal gives it an intentionally limited range. A ground-based system links the phone to only to the broadcast tower assigned to the specific small area, or cell, and smoothly transitions its connection to the next cell when the signal reception there is the stronger one. This enables a cell phone to be used while changing location.
Microwaves also provide the WiFi that enables owners of cell phones, laptop computers, and similar devices to connect wirelessly to the Internet at home and at coffee shops and airports. A wireless WiFi router is a device that exchanges data over the Internet through the cable or another connection, and uses microwaves to exchange the data wirelessly with devices such as cell phones and computers. The term WiFi itself refers to the standards followed in modulating and analyzing the microwaves so that wireless routers and devices from different manufacturers work compatibly with one another. The computer data in each direction consist of sequences of binary zeros and ones, each corresponding to a binary bit. The microwaves are in the range of 2.4 GHz to 5.0 GHz range.
Other wireless technologies also use microwaves to provide everyday communications between devices. Bluetooth developed alongside WiFi as a standard for radio communication in the 2.4-GHz range between nearby devices, for example, to link to headphones and audio earpieces to devices such as radios, or a driver’s cell phone to a hands-free device to allow answering phone calls without fumbling directly with the cell phone.
Microwaves find use also in radio tagging, using RFID (radio frequency identification) technology. Examples are RFID tags attached to store merchandize, transponder for toll booths use attached to the windshield of a car, or even a chip embedded into a pet’s skin. The device responds to a microwave signal by emitting a signal of its own with encoded information, allowing stores to quickly ring up items at their cash registers, drivers to charge tolls to their account without stopping, and lost pets to be reunited with their owners. NFC (near field communication) works similarly, except it is much shorter range. Its mechanism of interaction is the induced magnetic field at microwave frequencies between two coils. Cell phones that have NFC capability and the right software can supply information for purchases using the cell phone instead of a physical credit card. The very short range of the data transfer is a desired security feature in this case.
Infrared Radiation
The boundary between the microwave and infrared regions of the electromagnetic spectrum is not well defined (Figure \(\PageIndex{1}\)). Infrared radiation is generally produced by thermal motion, and the vibration and rotation of atoms and molecules. Electronic transitions in atoms and molecules can also produce infrared radiation . About half of the solar energy arriving at Earth is in the infrared region, with most of the rest in the visible part of the spectrum. About 23% of the solar energy is absorbed in the atmosphere, about 48% is absorbed at Earth’s surface, and about 29% is reflected back into space.
The range of infrared frequencies extends up to the lower limit of visible light, just below red. In fact, infrared means “below red.” Water molecules rotate and vibrate particularly well at infrared frequencies. Reconnaissance satellites can detect buildings, vehicles, and even individual humans by their infrared emissions, whose power radiation is proportional to the fourth power of the absolute temperature. More mundanely, we use infrared lamps, including those called quartz heaters , to preferentially warm us because we absorb infrared better than our surroundings.
The familiar handheld “remotes” for changing channels and settings on television sets often transmit their signal by modulating an infrared beam. If you try to use a TV remote without the infrared emitter being in direct line of sight with the infrared detector, you may find the television not responding. Some remotes use Bluetooth instead and reduce this annoyance.
Visible Light
Visible light is the narrow segment of the electromagnetic spectrum between about 400 nm and about 750 nm to which the normal human eye responds. Visible light is produced by vibrations and rotations of atoms and molecules, as well as by electronic transitions within atoms and molecules. The receivers or detectors of light largely utilize electronic transitions.
Red light has the lowest frequencies and longest wavelengths, whereas violet has the highest frequencies and shortest wavelengths (Figure \(\PageIndex{4}\)). Blackbody radiation from the Sun peaks in the visible part of the spectrum but is more intense in the red than in the violet, making the sun yellowish in appearance.
Living things - plants and animals - have evolved to utilize and respond to parts of the electromagnetic spectrum in which they are embedded. We enjoy the beauty of nature through visible light. Plants are more selective. Photosynthesis uses parts of the visible spectrum to make sugars.
Ultraviolet Radiation
Ultraviolet means “above violet.” The electromagnetic frequencies of ultraviolet radiation (UV) extend upward from violet, the highest-frequency visible light. The highest-frequency ultraviolet overlaps with the lowest-frequency X-rays. The wavelengths of ultraviolet extend from 400 nm down to about 10 nm at its highest frequencies. Ultraviolet is produced by atomic and molecular motions and electronic transitions.
UV radiation from the Sun is broadly subdivided into three wavelength ranges: UV-A (320–400 nm) is the lowest frequency, then UV-B (290–320 nm) and UV-C (220–290 nm). Most UV-B and all UV-C are absorbed by ozone (\(O_3)\) molecules in the upper atmosphere. Consequently, 99% of the solar UV radiation reaching Earth’s surface is UV-A.
Sunburn is caused by large exposures to UV-B and UV-C, and repeated exposure can increase the likelihood of skin cancer. The tanning response is a defense mechanism in which the body produces pigments in inert skin layers to reduce exposure of the living cells below.
As examined in a later chapter, the shorter the wavelength of light, the greater the energy change of an atom or molecule that absorbs the light in an electronic transition. This makes short-wavelength ultraviolet light damaging to living cells. It also explains why ultraviolet radiation is better able than visible light to cause some materials to glow, or fluoresce .
Besides the adverse effects of ultraviolet radiation, there are also benefits of exposure in nature and uses in technology. Vitamin D production in the skin results from exposure to UV-B radiation, generally from sunlight. Several studies suggest vitamin D deficiency is associated with the development of a range of cancers (prostate, breast, colon), as well as osteoporosis. Low-intensity ultraviolet has applications such as providing the energy to cause certain dyes to fluoresce and emit visible light, for example, in printed money to display hidden watermarks as counterfeit protection.
X-Rays
X-ray s have wavelengths from about \(10^{-8} m\) to \(10^{-12}m\). They have shorter wavelengths, and higher frequencies, than ultraviolet, so that the energy they transfer at an atomic level is greater. As a result, X-rays have adverse effects on living cells similar to those of ultraviolet radiation, but they are more penetrating. Cancer and genetic defects can be induced by X-rays. Because of their effect on rapidly dividing cells, X-rays can also be used to treat and even cure cancer.
The widest use of X-rays is for imaging objects that are opaque to visible light, such as the human body or aircraft parts. In humans, the risk of cell damage is weighed carefully against the benefit of the diagnostic information obtained.
Gamma Rays
Soon after nuclear radioactivity was first detected in 1896, it was found that at least three distinct types of radiation were being emitted, and these were designated as alpha, beta, and gamma rays. The most penetrating nuclear radiation, the gamma ray \(\gamma\) -ray) was later found to be an extremely high-frequency electromagnetic wave.
The lower end of the \(\gamma\)-ray frequency range overlaps the upper end of the X-ray range. Gamma rays have characteristics identical to X-rays of the same frequency—they differ only in source. The name “gamma rays” is generally used for electromagnetic radiation emitted by a nucleus, while X-rays are generally produced by bombarding a target with energetic electrons in an X-ray tube. At higher frequencies, \(\gamma\)-rays are more penetrating and more damaging to living tissue. They have many of the same uses as X-rays, including cancer therapy. Gamma radiation from radioactive materials is used in nuclear medicine.
Use this simulation to explore how light interacts with molecules in our atmosphere.
- Explore how light interacts with molecules in our atmosphere.
- Identify that absorption of light depends on the molecule and the type of light.
- Relate the energy of the light to the resulting motion.
- Identify that energy increases from microwave to ultraviolet.
- Predict the motion of a molecule based on the type of light it absorbs.
Exercise \(\PageIndex{1}\)
How do the electromagnetic waves for the different kinds of electromagnetic radiation differ?
- Answer
-
They fall into different ranges of wavelength, and therefore also different corresponding ranges of frequency.


4.7: Personality Theory in Real Life
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Achieving Athletic Excellence Despite Physical Challenges
Adler’s studies on inferiority began with physical problems, what he called organ inferiority (Adler, 1917). Most students of Adler look past that medical beginning, and focus instead on the psychological inferiorities that children experience during their development. However, there are many people with organ inferiority, or what we more commonly refer to as disabilities, handicaps, or “challenges.” There may be some debate as to which term is preferred, but since the phrase “politically correct” is itself a contradiction in terms, I will use the terms disability and handicap as presented in Warren Rule’s book Lifestyle Counseling for Adjustment to Disability (Rule, 1984). In his summary of previous research, Rule adopts the definition of a disability as a “relatively severe chronic impairment of function” that occurs as the result of a congenital defect, disease, or an accident. Accordingly, disability refers to actual physical, mental, or emotional impairments that become a handicap only if they cause lowered self-assessment, reduced activity, or limited opportunities. When disabilities become a handicap, they can affect the individual’s entire style of life. Thus, Rule brought together a group of therapists trained in Individual Psychology, and published the aforementioned book on using lifestyle counseling for people with disabilities that have led to handicaps.
However, not everyone with a disability develops a handicap. Instead, some individuals become truly inspirational by the way in which they live their lives in spite of their disability, or rather, as if they simply were not disabled. Erik Weihenmayer (2001; see also Stoltz & Weihenmayer, 2006) was born with retinoscheses, a degenerative eye disease, which slowly destroyed his retinas, leaving him blind by the age of 13. In high school, Erik spent a month one summer at the Carroll Center for the Blind in Massachusetts. The summer camp included a weekend of rock climbing in N. Conway, New Hampshire (where the author has done a lot of rock-climbing). Weihenmayer’s rock climbing experience altered his life. He continued climbing rock, and then moved on to ice-climbing and mountaineering. He didn’t just followed more experienced climbers up the cliffs, he also learned to lead-climb: placing one’s own protection along the climb and then clipping in the rope, what climbers call “the sharp end” of the rope. I had the pleasure of climbing with Erik in Michigan’s Upper Peninsula a few years ago, while his seeing-eye dog slept in a sort of ice cave formed by the overhanging ice. It is truly extraordinary to watch him climb. He moves so smoothly, as he feels the ice above with his ice axe, and then sets the ice axe so deliberately when he finds the right spot, that you would not know he was climbing blind if you only watched for a little while. Eventually, Erik decided to pursue the Seven Summits, climbing the highest peak on each continent: Mt. McKinley (N. America), Aconcagua (S. America), Mt. Everest (Asia), Mt. Elbrus (Europe), Vinson Massif (Antarctica), Mt. Kosciusko (Australia), Kilimanjaro (Africa). He accomplished his goal in 2002.
Erik Weihenmayer is by no means the only well-known, disabled climber. In an amazing video, Beyond the Barriers (Perlman & Wellman, 1998), Erik goes climbing with Mark Wellman and Hugh Herr. Wellman was paralyzed from the waist down in a climbing accident (Wellman & Flinn, 1992), and Herr lost both of his lower legs to frostbite after being caught in a vicious winter storm on Mt. Washington, NH (Osius, 1991; Note: The author has suffered a small patch of frostbite during a winter storm on Mt. Washington). In Beyond the Barriers, Herr leads the hike toward the climb, while Erik carries Wellman. Once on the climb, Herr leads the climb, Erik follows, and they set ropes for Wellman to do pull-ups up the cliff. It simply has to be seen to be believed. One of the surprising aspects is how they joke with each other about what they are doing. As Erik is carrying Wellman, Wellman says: “I don’t know man. A blind man giving a para a piggyback ride? It’s a pretty scary thing!” When Herr starts climbing on a day when it was snowing, he says his hands are getting numb from the cold. So, Erik asks him how his feet feel! Humor was always an important part of Adlerian psychotherapy (Scott, 1984), so perhaps it should not be surprising that a sense of humor is an aspect of their personalities. One of the funniest stories that Erik tells is about the time he accidentally drank out of his climbing partners piss bottle (a bottle used to urinate inside the tent during storms). Erik became quite upset that the bottle wasn’t marked somehow, but his partner defended himself by saying he had clearly written on the bottle which one it was. It slowly dawned on Erik’s partner that the writing was of no help to Erik. As another example of Erik’s humor, consider the challenge he tried to avoid after having climbed the highest peaks in Africa and North and South America:
Emma Louise Weihenmayer was born on June 21, year 2000, at 3:57 A.M. There is so much to learn about parenthood. Sometimes being a father is about as intense as climbing Denali, Kilimanjaro, and Aconcagua, all in a day. Because I’m blind, I tried to convince Ellie that I couldn’t change diapers, but for some reason, she didn’t buy it. (pg. 303; Weihenmayer, 2001)
In addition to his climbing, Erik Weihenmayer is a college graduate with a teaching certificate, and he spent some time as a middle school teacher. He also tried the sport of wrestling, and was a wrestling coach. Trevon Jenifer was also a wrestler.
Trevon Jenifer was born without legs. Perhaps even more challenging, however, was the fact that he was the fourth child of a poor, single mother living in a ghetto outside of Washington, DC. Obviously, Trey (the name he goes by) began life facing difficult obstacles, but little by little, things got better. His mother, Connie, made a conscious decision to take care of him the best she could.  He met a wonderful special education teacher named Bob Gray, who got Trey interested in sports, and who helped to make participating in sports a realistic possibility. He eventually joined a wheelchair track and basketball team named Air Capital, and he was very successful on the track, setting national records in the 100-, 200-, and 400-meter races. It was prior to his junior year in high school, however, that his step-father, who had been a wrestler, recommended that Trey try out for the wrestling team, the regular wrestling team.
What Trey wanted more than anything was to fit in, to have a normal social life at school. Being in a wheelchair, that was not likely to happen. However, he felt that sports might help him accomplish that goal, so he did try out for the wrestling team. He worked hard, learned as much as he could, and he made the varsity team as the 103-pound competitor (actually, there was no one else that light on the team, but he didn’t know that). His coach, Terry Green, did all he could to help Trey find a wrestling style that would take advantage of his relative arm strength (he made weight without legs, so his upper body was relatively large) while overcoming the disadvantage of not being able to balance or leverage his body weight by spreading out his legs. Now it was up to Trey. He was nervous in his first match, didn’t assert himself, and was easily pinned. In his second match he became the aggressor and earned his first victory. The rest of his junior year continued to be a series of wins and losses, and he ended the season 17-18. Of course, it had only been his first season of wrestling.
In anticipation of his senior year in high school, Trey continued to train hard. Outside of the ring he also received recognition, and became a part of the social network of the school. He received a Medal of Courage from the National Wrestling Hall of Fame, he attended his school’s prom, and he was chosen as co-captain of the wrestling team. Once again, humor played a role, as he compared his strength to a teammate from the previous year. Trey had made significant strides in how much he could bench-press, so his former teammate asked him how much he could squat (a lift done entirely with the legs)! Both wrestlers enjoyed a good laugh at that one.
Trey was doing quite well in league wrestling, and he also began to do well in tournaments. Eventually, he won a tournament, ended his season at 26-6, and from there went on the state championship. He won his first match, but then had to face an undefeated wrestler. He lost, but in that loss there was a sense of accomplishment due to how far he had come:
I lost 5-2…I was hurt less by the fact that I lost, and more by the closeness of it. This one hurt even more because of how close I was to beating the best wrestler in the state. Sharbaugh went on to win the state championship. In fact, he won his last two matches very convincingly, 6-0, and, 12-5. He told reporters afterward that my match was his toughest of the tournament. (pp. 171-172; Trevon Jenifer in Jenifer & Goldenbach, 2006).
The next morning he had to return to the championships to wrestle for a chance at third place in the state. He began with a vengeance, scoring victories of 9-1 and 9-2. His next match, and a shot at third place, was not so easy, but he won 3-1, earning his 30 th victory of the season. He then won his final match, and earned third place in the state championship. However, an even more important challenge now loomed ahead of him: college.
Coming from a poor, Black family, there was no tradition of children going to college. However, a group of concerned philanthropists became interested in supporting his dreams. His old coach at Air Capital had talked to Jim Glatch, who coached wheelchair basketball at Edinboro College in Pennsylvania, a school with a large population (10 percent) of students with disabilities. Trevon Jenifer currently attends Edinboro College and plays on the wheelchair basketball team. He does not know if he will ever wrestle again, but it is interesting to note that Edinboro College has a famous wrestler as their athletic director: two-time Olympic gold medalist Bruce Baumgartner!
Recently, Trey was kind enough to respond to an email I sent him, and he provided me with an update on how things have gone during his first year of college. He misses wrestling very much, but he has really enjoyed his return to wheelchair basketball. It probably didn’t hurt that the team is very good, and they came in second-place in the NCAA championship for wheelchair basketball. Trey maintained good grades, his family strongly supports him in pursuing his education, and he has made many new friends. But a few challenges remain. It has been a little difficult for him to get used to the weather in northwestern Pennsylvania, and he has been too busy to attend as many book signings as his publisher would like (but he says they have been very understanding). As for becoming the inspiration his mother thought he was born to be:
I think that I have inspired some people, and I think that is great, but I don’t think that it has reached all the people that I would like it too. My family says that I have done a good job, but I [think that I could do a lot better], and I will try until I feel that I have reached that. (Trevon Jenifer; personal communication, 2007)
The range of sports in which disabled individuals compete is extraordinary today. Beyond the Barriers also includes disabled individuals sailing, scuba diving, surfing, and hang gliding. A few years ago, I began practicing Tae Kwon Do, and I soon discovered that I had degenerative joint disease in both hips. I considered quitting Tae Kwon Do, but I was strongly encouraged to continue by my instructors, as well as by my orthopedic surgeon and physical therapist. I have since learned that Dirk Robertson, a former social worker turned actor and writer, has worked hard promoting martial arts training for people with disabilities (Robertson, 1991; see also McNab, 2003). Each person simply needs to be encouraged to do their best. Adler suggested that the best way to strive for superiority was through social interest. Whether it’s a climbing partner, a wrestling team, a wheelchair basketball team, a martial arts school, whatever, when people work together to help each individual achieve their potential, it can prove to be a highly rewarding experience.
Whilst it is important to be sensitive to their particular situation, their disability should not be the central focus all the time. Their ability to learn, listen and adapt should be built on and encouraged. Do not be over-protective or an instant expert on people with disabilities. The experts are the people themselves, so listen to what they have to say. (pp. 101-102; Robertson, 1991)


Adverbs
are words that modify or describe a verb, adjective, or another adverb. Just as an adjective changes a noun, an adverb changes a verb, adjective, or adverb. Adverbs are easily identified because they often end in
ly
, but this is certainly not always the case.
Descriptions make our writing rich and specific, so we shouldn’t be afraid of using adjectives and adverbs in our sentences.
Look at these three sentences:
example
Jon
walked to the store to get canned goods for his zombie stash.
Jon walked to the large store to get canned goods for his zombie stash.
Jon walked
urgently
to the
massively
large store to get canned goods for his zombie stash.
As you can see, the last sentence is the most descriptive and informative. The use of adverbs and adjectives helps our writing come alive. The following page on adverbs will provide helpful tips on how to correctly use adverbs and give you examples of how adverbs can add descriptive detail to your writing.
Order of Adverbs
Adverbs most commonly describe how, but below is a more comprehensive list of the most common types of adverbs.
|
Type of Adverb
|
Example
|
|
Adverbs of manner (or how)
|
Christine sang the song
atrociously
. No more karaoke for her!
|
|
Adverbs of time
|
Michelle did her homework
yesterday
, but she did the wrong assignment.
|
|
Adverbs of place
|
I met my friend
at the coffee shop
, and that’s where we saw the first signs of the outbreak.
|
|
Adverbs of degree
|
It’s
too
quiet in here.
|
|
Adverbs of frequency
|
Michael Jordan
rarely
misses a free throw, but Shaq
frequently
does.
|
|
Adverbs of purpose
|
I clean the litter box every day
to keep the house from smelling
.
|
And like adjectives, adverbs have a “royal order.” While you may already have an innate sense of this order, it can be helpful to review the rules.
The Royal Order of Adverbs
Source: Adapted from Adverbs. (n.d.) Capital Community College Foundation. Retrieved from grammar.ccc.commnet.edu


4.4: Practice Time
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Exercise 1. Look back at Exercise 3 in Chapter 4. Given what you concluded about the phonology of Japanese in that exercise, predict whether it would likely be easy or difficult for an L1 speaker of Japanese to learn the English contrast between words like sip/ship , sign/shine , class/clash . Explain how you arrived at your conclusion.
Exercise 2. Look again at Exercise 2 in Chapter 4. In that exercise, you concluded that German has a phoneme [ʃ] that contrasts with another phoneme that has two allophones, [x] and [ç] (a voiceless velar fricative and voiceless palatal fricative, respectively). Given what you know about English phonology, how easy or difficult will it be for an L1 speaker of English to learn this contrast in German? Explain how you arrived at your conclusion.