3.7: Eukaryotic External Structures
-
- Last updated
- Save as PDF
Skills to Develop
- Describe the structure of eukaryotic cells
- Compare animal cells with plant cells
- State the role of the plasma membrane
- Summarize the functions of the major cell organelles
Have you ever heard the phrase “form follows function?” It’s a philosophy practiced in many industries. In architecture, this means that buildings should be constructed to support the activities that will be carried out inside them. For example, a skyscraper should be built with several elevator banks; a hospital should be built so that its emergency room is easily accessible.
Our natural world also utilizes the principle of form following function, especially in cell biology, and this will become clear as we explore eukaryotic cells (Figure \(\PageIndex{1}\)). Unlike prokaryotic cells, eukaryotic cells have: 1) a membrane-bound nucleus; 2) numerous membrane-bound organelles such as the endoplasmic reticulum, Golgi apparatus, chloroplasts, mitochondria, and others; and 3) several, rod-shaped chromosomes. Because a eukaryotic cell’s nucleus is surrounded by a membrane, it is often said to have a “true nucleus.” The word “organelle” means “little organ,” and, as already mentioned, organelles have specialized cellular functions, just as the organs of your body have specialized functions.
At this point, it should be clear to you that eukaryotic cells have a more complex structure than prokaryotic cells. Organelles allow different functions to be compartmentalized in different areas of the cell. Before turning to organelles, let’s first examine two important components of the cell: the plasma membrane and the cytoplasm.
Exercise \(\PageIndex{1}\)
If the nucleolus were not able to carry out its function, what other cellular organelles would be affected?
The Plasma Membrane
Like prokaryotes, eukaryotic cells have a plasma membrane (Figure \(\PageIndex{2}\)), a phospholipid bilayer with embedded proteins that separates the internal contents of the cell from its surrounding environment.  The plasma membrane controls the passage of organic molecules, ions, water, and oxygen into and out of the cell. Wastes (such as carbon dioxide and ammonia) also leave the cell by passing through the plasma membrane.
The plasma membranes of cells that specialize in absorption are folded into fingerlike projections called microvilli (singular = microvillus); (Figure \(\PageIndex{3}\)). Such cells are typically found lining the small intestine, the organ that absorbs nutrients from digested food. This is an excellent example of form following function. People with celiac disease have an immune response to gluten, which is a protein found in wheat, barley, and rye. The immune response damages microvilli, and thus, afflicted individuals cannot absorb nutrients. This leads to malnutrition, cramping, and diarrhea. Patients suffering from celiac disease must follow a gluten-free diet.
The Cytoplasm
The cytoplasm is the entire region of a cell between the plasma membrane and the nuclear envelope (a structure to be discussed shortly). It is made up of organelles suspended in the gel-like cytosol , the cytoskeleton, and various chemicals (Figure \(\PageIndex{1}\)). Even though the cytoplasm consists of 70 to 80 percent water, it has a semi-solid consistency, which comes from the proteins within it. However, proteins are not the only organic molecules found in the cytoplasm. Glucose and other simple sugars, polysaccharides, amino acids, nucleic acids, fatty acids, and derivatives of glycerol are found there, too. Ions of sodium, potassium, calcium, and many other elements are also dissolved in the cytoplasm. Many metabolic reactions, including protein synthesis, take place in the cytoplasm.
The Nucleus
Typically, the nucleus is the most prominent organelle in a cell (Figure \(\PageIndex{1}\)). The nucleus (plural = nuclei) houses the cell’s DNA and directs the synthesis of ribosomes and proteins. Let’s look at it in more detail (Figure \(\PageIndex{4}\)).
The Nuclear Envelope
The nuclear envelope is a double-membrane structure that constitutes the outermost portion of the nucleus (Figure \(\PageIndex{4}\)). Both the inner and outer membranes of the nuclear envelope are phospholipid bilayers.
The nuclear envelope is punctuated with pores that control the passage of ions, molecules, and RNA between the nucleoplasm and cytoplasm. The nucleoplasm is the semi-solid fluid inside the nucleus, where we find the chromatin and the nucleolus.
Chromatin and Chromosomes
To understand chromatin, it is helpful to first consider chromosomes. Chromosomes are structures within the nucleus that are made up of DNA, the hereditary material. You may remember that in prokaryotes, DNA is organized into a single circular chromosome. In eukaryotes, chromosomes are linear structures. Every eukaryotic species has a specific number of chromosomes in the nuclei of its body’s cells. For example, in humans, the chromosome number is 46, while in fruit flies, it is eight. Chromosomes are only visible and distinguishable from one another when the cell is getting ready to divide. When the cell is in the growth and maintenance phases of its life cycle, proteins are attached to chromosomes, and they resemble an unwound, jumbled bunch of threads. These unwound protein-chromosome complexes are called chromatin (Figure \(\PageIndex{5}\)); chromatin describes the material that makes up the chromosomes both when condensed and decondensed.
The Nucleolus
We already know that the nucleus directs the synthesis of ribosomes, but how does it do this? Some chromosomes have sections of DNA that encode ribosomal RNA. A darkly staining area within the nucleus called the nucleolus (plural = nucleoli) aggregates the ribosomal RNA with associated proteins to assemble the ribosomal subunits that are then transported out through the pores in the nuclear envelope to the cytoplasm.
Ribosomes
Ribosomes are the cellular structures responsible for protein synthesis. When viewed through an electron microscope, ribosomes appear either as clusters (polyribosomes) or single, tiny dots that float freely in the cytoplasm. They may be attached to the cytoplasmic side of the plasma membrane or the cytoplasmic side of the endoplasmic reticulum and the outer membrane of the nuclear envelope (Figure \(\PageIndex{1}\)). Electron microscopy has shown us that ribosomes, which are large complexes of protein and RNA, consist of two subunits, aptly called large and small (Figure \(\PageIndex{6}\)). Ribosomes receive their “orders” for protein synthesis from the nucleus where the DNA is transcribed into messenger RNA (mRNA). The mRNA travels to the ribosomes, which translate the code provided by the sequence of the nitrogenous bases in the mRNA into a specific order of amino acids in a protein. Amino acids are the building blocks of proteins.
Because proteins synthesis is an essential function of all cells (including enzymes, hormones, antibodies, pigments, structural components, and surface receptors), ribosomes are found in practically every cell. Ribosomes are particularly abundant in cells that synthesize large amounts of protein. For example, the pancreas is responsible for creating several digestive enzymes and the cells that produce these enzymes contain many ribosomes. Thus, we see another example of form following function.
Mitochondria
Mitochondria (singular = mitochondrion) are often called the “powerhouses” or “energy factories” of a cell because they are responsible for making adenosine triphosphate (ATP), the cell’s main energy-carrying molecule. ATP represents the short-term stored energy of the cell. Cellular respiration is the process of making ATP using the chemical energy found in glucose and other nutrients. In mitochondria, this process uses oxygen and produces carbon dioxide as a waste product.
In keeping with our theme of form following function, it is important to point out that muscle cells have a very high concentration of mitochondria that produce ATP. Your muscle cells need a lot of energy to keep your body moving. When your cells don’t get enough oxygen, they do not make a lot of ATP. Instead, the small amount of ATP they make in the absence of oxygen is accompanied by the production of lactic acid.
Mitochondria are oval-shaped, double membrane organelles (Figure \(\PageIndex{7}\)) that have their own ribosomes and DNA. Each membrane is a phospholipid bilayer embedded with proteins. The inner layer has folds called cristae. The area surrounded by the folds is called the mitochondrial matrix. The cristae and the matrix have different roles in cellular respiration.
Peroxisomes
Peroxisomes are small, round organelles enclosed by single membranes. They carry out oxidation reactions that break down fatty acids and amino acids. They also detoxify many poisons that may enter the body. (Many of these oxidation reactions release hydrogen peroxide, H 2 O 2 , which would be damaging to cells; however, when these reactions are confined to peroxisomes, enzymes safely break down the H 2 O 2 into oxygen and water.) For example, alcohol is detoxified by peroxisomes in liver cells. Glyoxysomes, which are specialized peroxisomes in plants, are responsible for converting stored fats into sugars.
Vesicles and Vacuoles
Vesicles and vacuoles are membrane-bound sacs that function in storage and transport. Other than the fact that vacuoles are somewhat larger than vesicles, there is a very subtle distinction between them: The membranes of vesicles can fuse with either the plasma membrane or other membrane systems within the cell. Additionally, some agents such as enzymes within plant vacuoles break down macromolecules. The membrane of a vacuole does not fuse with the membranes of other cellular components.
Animal Cells versus Plant Cells
At this point, you know that each eukaryotic cell has a plasma membrane, cytoplasm, a nucleus, ribosomes, mitochondria, peroxisomes, and in some, vacuoles, but there are some striking differences between animal and plant cells. While both animal and plant cells have microtubule organizing centers (MTOCs), animal cells also have centrioles associated with the MTOC: a complex called the centrosome. Animal cells each have a centrosome and lysosomes, whereas plant cells do not. Plant cells have a cell wall, chloroplasts and other specialized plastids, and a large central vacuole, whereas animal cells do not.
The Centrosome
The centrosome is a microtubule-organizing center found near the nuclei of animal cells. It contains a pair of centrioles, two structures that lie perpendicular to each other (Figure \(\PageIndex{8}\)). Each centriole is a cylinder of nine triplets of microtubules.
The centrosome (the organelle where all microtubules originate) replicates itself before a cell divides, and the centrioles appear to have some role in pulling the duplicated chromosomes to opposite ends of the dividing cell. However, the exact function of the centrioles in cell division isn’t clear, because cells that have had the centrosome removed can still divide, and plant cells, which lack centrosomes, are capable of cell division.
Lysosomes
Animal cells have another set of organelles not found in plant cells: lysosomes. The lysosomes are the cell’s “garbage disposal.” In plant cells, the digestive processes take place in vacuoles. Enzymes within the lysosomes aid the breakdown of proteins, polysaccharides, lipids, nucleic acids, and even worn-out organelles. These enzymes are active at a much lower pH than that of the cytoplasm. Therefore, the pH within lysosomes is more acidic than the pH of the cytoplasm. Many reactions that take place in the cytoplasm could not occur at a low pH, so again, the advantage of compartmentalizing the eukaryotic cell into organelles is apparent.
The Cell Wall
If you examine Figure \(\PageIndex{1}\)b, the diagram of a plant cell, you will see a structure external to the plasma membrane called the cell wall. The cell wall is a rigid covering that protects the cell, provides structural support, and gives shape to the cell. Fungal and protistan cells also have cell walls. While the chief component of prokaryotic cell walls is peptidoglycan, the major organic molecule in the plant cell wall is cellulose (Figure \(\PageIndex{9}\)), a polysaccharide made up of glucose units. Have you ever noticed that when you bite into a raw vegetable, like celery, it crunches? That’s because you are tearing the rigid cell walls of the celery cells with your teeth.
Chloroplasts
Like the mitochondria, chloroplasts have their own DNA and ribosomes, but chloroplasts have an entirely different function. Chloroplasts are plant cell organelles that carry out photosynthesis. Photosynthesis is the series of reactions that use carbon dioxide, water, and light energy to make glucose and oxygen. This is a major difference between plants and animals; plants (autotrophs) are able to make their own food, like sugars, while animals (heterotrophs) must ingest their food.
Like mitochondria, chloroplasts have outer and inner membranes, but within the space enclosed by a chloroplast’s inner membrane is a set of interconnected and stacked fluid-filled membrane sacs called thylakoids (Figure \(\PageIndex{10}\)). Each stack of thylakoids is called a granum (plural = grana). The fluid enclosed by the inner membrane that surrounds the grana is called the stroma.
The chloroplasts contain a green pigment called chlorophyll , which captures the light energy that drives the reactions of photosynthesis. Like plant cells, photosynthetic protists also have chloroplasts. Some bacteria perform photosynthesis, but their chlorophyll is not relegated to an organelle.
Evolution Connection: Endosymbiosis
We have mentioned that both mitochondria and chloroplasts contain DNA and ribosomes. Have you wondered why? Strong evidence points to endosymbiosis as the explanation.
Symbiosis is a relationship in which organisms from two separate species depend on each other for their survival. Endosymbiosis (endo- = “within”) is a mutually beneficial relationship in which one organism lives inside the other. Endosymbiotic relationships abound in nature. We have already mentioned that microbes that produce vitamin K live inside the human gut. This relationship is beneficial for us because we are unable to synthesize vitamin K. It is also beneficial for the microbes because they are protected from other organisms and from drying out, and they receive abundant food from the environment of the large intestine.
Scientists have long noticed that bacteria, mitochondria, and chloroplasts are similar in size. We also know that bacteria have DNA and ribosomes, just as mitochondria and chloroplasts do. Scientists believe that host cells and bacteria formed an endosymbiotic relationship when the host cells ingested both aerobic and autotrophic bacteria (cyanobacteria) but did not destroy them. Through many millions of years of evolution, these ingested bacteria became more specialized in their functions, with the aerobic bacteria becoming mitochondria and the autotrophic bacteria becoming chloroplasts.
The Central Vacuole
Previously, we mentioned vacuoles as essential components of plant cells. If you look at Figure \(\PageIndex{1}\)b, you will see that plant cells each have a large central vacuole that occupies most of the area of the cell. The central vacuole plays a key role in regulating the cell’s concentration of water in changing environmental conditions. Have you ever noticed that if you forget to water a plant for a few days, it wilts? That’s because as the water concentration in the soil becomes lower than the water concentration in the plant, water moves out of the central vacuoles and cytoplasm. As the central vacuole shrinks, it leaves the cell wall unsupported. This loss of support to the cell walls of plant cells results in the wilted appearance of the plant.
The central vacuole also supports the expansion of the cell. When the central vacuole holds more water, the cell gets larger without having to invest a lot of energy in synthesizing new cytoplasm.
Summary
Like a prokaryotic cell, a eukaryotic cell has a plasma membrane, cytoplasm, and ribosomes, but a eukaryotic cell is typically larger than a prokaryotic cell, has a true nucleus (meaning its DNA is surrounded by a membrane), and has other membrane-bound organelles that allow for compartmentalization of functions. The plasma membrane is a phospholipid bilayer embedded with proteins. The nucleus’s nucleolus is the site of ribosome assembly. Ribosomes are either found in the cytoplasm or attached to the cytoplasmic side of the plasma membrane or endoplasmic reticulum. They perform protein synthesis. Mitochondria participate in cellular respiration; they are responsible for the majority of ATP produced in the cell. Peroxisomes hydrolyze fatty acids, amino acids, and some toxins. Vesicles and vacuoles are storage and transport compartments. In plant cells, vacuoles also help break down macromolecules.
Animal cells also have a centrosome and lysosomes. The centrosome has two bodies perpendicular to each other, the centrioles, and has an unknown purpose in cell division. Lysosomes are the digestive organelles of animal cells.
Plant cells and plant-like cells each have a cell wall, chloroplasts, and a central vacuole. The plant cell wall, whose primary component is cellulose, protects the cell, provides structural support, and gives shape to the cell. Photosynthesis takes place in chloroplasts. The central vacuole can expand without having to produce more cytoplasm.
Art Connections
Figure \(\PageIndex{1}\): If the nucleolus were not able to carry out its function, what other cellular organelles would be affected?
- Answer
-
Free ribosomes and rough endoplasmic reticulum (which contains ribosomes) would not be able to form.
Glossary
- cell wall
- rigid cell covering made of cellulose that protects the cell, provides structural support, and gives shape to the cell
- central vacuole
- large plant cell organelle that regulates the cell’s storage compartment, holds water, and plays a significant role in cell growth as the site of macromolecule degradation
- centrosome
- region in animal cells made of two centrioles
- chlorophyll
- green pigment that captures the light energy that drives the light reactions of photosynthesis
- chloroplast
- plant cell organelle that carries out photosynthesis
- chromatin
- protein-DNA complex that serves as the building material of chromosomes
- chromosome
- structure within the nucleus that is made up of chromatin that contains DNA, the hereditary material
- cytoplasm
- entire region between the plasma membrane and the nuclear envelope, consisting of organelles suspended in the gel-like cytosol, the cytoskeleton, and various chemicals
- cytosol
- gel-like material of the cytoplasm in which cell structures are suspended
- eukaryotic cell
- cell that has a membrane-bound nucleus and several other membrane-bound compartments or sacs
- lysosome
- organelle in an animal cell that functions as the cell’s digestive component; it breaks down proteins, polysaccharides, lipids, nucleic acids, and even worn-out organelles
- mitochondria
- (singular = mitochondrion) cellular organelles responsible for carrying out cellular respiration, resulting in the production of ATP, the cell’s main energy-carrying molecule
- nuclear envelope
- double-membrane structure that constitutes the outermost portion of the nucleus
- nucleolus
- darkly staining body within the nucleus that is responsible for assembling the subunits of the ribosomes
- nucleoplasm
- semi-solid fluid inside the nucleus that contains the chromatin and nucleolus
- nucleus
- cell organelle that houses the cell’s DNA and directs the synthesis of ribosomes and proteins
- organelle
- compartment or sac within a cell
- peroxisome
- small, round organelle that contains hydrogen peroxide, oxidizes fatty acids and amino acids, and detoxifies many poisons
- plasma membrane
- phospholipid bilayer with embedded (integral) or attached (peripheral) proteins, and separates the internal content of the cell from its surrounding environment
- ribosome
- cellular structure that carries out protein synthesis
- vacuole
- membrane-bound sac, somewhat larger than a vesicle, which functions in cellular storage and transport
- vesicle
- small, membrane-bound sac that functions in cellular storage and transport; its membrane is capable of fusing with the plasma membrane and the membranes of the endoplasmic reticulum and Golgi apparatus


9.0: Introduction
Neurological disorders affect the body’s autonomic, peripheral, and central nervous system. These disorders may occur at different points in the lifespan of an individual and for a variety of reasons. The severity of neurological disorders also varies greatly, with some disorders being mild and self-limiting to other disorders that may be quite debilitating or even life-threatening. Some examples of neurological disorders are migraines, non-migraine headaches, multiple sclerosis, dementia such as Alzheimer’s disease, Parkinson’s disease, brain tumors, spinal cord injuries, stroke, and epilepsy. Clients with a neurological disorder may experience a wide range of signs and symptoms. Therefore, a thorough neurological assessment is vital to detect deficits or changes in baseline neurological functioning. Signs and symptoms of a neurological disorder include headaches, memory or sensation loss, muscle weakness, tremors, nausea/vomiting, loss of bowel and/or bladder control, seizures, impacted speech, aphasia (trouble speaking), dysphagia (swallowing difficulties), intractable pain, blindness, and paralysis, all of which may vary in severity from mild to complete loss (Sargsyan, 2020; Shahrokhi & Asuncion, 2023). The occurrence of neurological disorders may be related to factors including genetics, infections, trauma, autoimmune reactions, or degenerative processes.
The Pan American Health Organization (PAHO) identified the overall incidence of noncommunicable neurological disorders across the Americas. Neurological disorders related to death and disability were recognized as a global public health challenge and are expected to continue increasing as the population ages. The United States is the highest ranked country from the region of the Americas for age-standardized death rates related to noncommunicable neurological disorders. Females were more affected (60%) than males (40%), but males had a slightly higher death rate compared to females (33.1 per 100,000 and 32.2 per 100,000, respectively) (PAHO, 2021). Nationally, treatment of neurological disorders is also recognized as a public health objective. The Healthy People 2030 objectives focus on various health conditions related to neurological disorders (i.e., dementia, stroke, sensory or communication disorders, and mental health disorders) (Office of Disease Prevention and Health Promotion, 2021).
Trending Today
Tracking the Epidemiology of Neurological Conditions
The National Neurological Conditions Surveillance System (NNCSS) (CDC, 2023) is a national surveillance system overseen by the Centers for Disease Control and Prevention (CDC) that tracks the epidemiology of neurological disorders. The aim of the NNCSS is to use the gathered epidemiological data to extend knowledge about neurological disorders and future research efforts to investigate related causes, diagnoses, and treatment.


20.1: Batteries and Simple Circuits
-
- Last updated
- Save as PDF
A battery is an electric component that provides a constant electric potential difference (a fixed voltage) across its terminals. Luigi Galvani was the first to realize that certain combination of metals placed into contact with each other can lead to an electric potential difference (or rather, they can make the legs of a dead frog twitch, which we now understand to be from the potential difference due to the metals). Effectively, Galvani created the first “electrochemical cell”. Alessandro Volta then combined several of these cells together to form the “voltaic pile”, which is what we would now call a battery (a battery, technically, is a combination of several cells, a battery of cells, although one often uses the term battery even if only a single electric cell is involved).
The Electrochemical Cell
An electric cell can be constructed from metals that have different affinities to be dissolved in acid. A simple cell, similar to that originally made by Volta, can be made using zinc and carbon as the “electrodes” (Volta used silver instead of carbon) and a solution of dilute sulfuric acid (the liquid is called the “electrolyte”), as illustrated in Figure \(\PageIndex{1}\). Before the cell is constructed, the electrodes and the electrolyte are all electrically neutral.
Once the zinc is immersed in the electrolyte, the zinc atoms tend to dissolve into the electrolyte in the form of zinc ions (doubly charged, Zn \(^{2+}\) ). This leaves an excess of electrons on the zinc electrode, resulting in a net negative electric charge. Similarly, the positively charged zinc ions attract electrons from the carbon electrode into the solution, leaving the carbon electrode positively charged. Very quickly, an equilibrium is reached, since at some point, the negative charge of the zinc electrode will electrically attract positive zinc ions, preventing any more zinc ions from dissolving into the solution. Similarly, as the carbon electrode builds a positive charge, that charge will eventually prevent electrons from “jumping” into the solution. At this point, there will be a fixed electric potential difference between the two electrodes (terminals) of the battery.
If the two electrodes are connected together through a resistor, the electrons will leave the zinc electrode, cross the resistor, and end up on the positive carbon electrode. This will leave space for more electrons on the zinc electrode, so more zinc ions will dissolve into the solution. Thus, a circuit is formed, where electron travel up the zinc electrode, through the resistor and back down the carbon electrode. At the same time, more and more zinc ions dissolve into the electrolyte, until the zinc electrode is completely dissolved. In practice, the zinc ions travel through the solution and plate onto the carbon electrode (the electrons do not quite “jump” into the electrolyte, rather, it is the zinc ions that move in the electrolyte). Since the charge on the electrodes is continuously replenished, the potential difference between the electrodes remains constant even as current is flowing.
The electric cell will stop working once the zinc electrode has completely dissolved (this is what happens when your battery is dead). Note that there is also a maximum current that the cell can supply, which depends on the rate at which the zinc can dissolve into the electrolyte and plate onto the carbon electrode. If the electrodes of the cell are connected with a very low resistance resistor, the resulting current will be too large for the potential difference to be maintained. Most electric cells work in similar ways, although the chemical reactions can be much more complex. Sometimes, the chemical reaction is reversible; one could use a different battery to apply a negative voltage to the carbon electrode to reverse the reaction and plate the zinc back onto the zinc electrode, thus “recharging the battery” (and converting electric energy back into stored chemical potential energy).
The ideal battery in a circuit
As we proceed, we will use the term “battery” loosely to refer to a device (such as an electric cell or collection of cells) that can provide a fixed potential difference between two terminals (or electrodes). Figure \(\PageIndex{2}\) shows the circuit diagram for a battery, consisting in two (or four) vertical bars, with the larger bar indicating the positive terminal of the battery.
Figure \(\PageIndex{3}\) shows the circuit diagram symbols that are used for a resistor (different symbols are used in North American and in Europe).
Figure \(\PageIndex{4}\) shows a circuit diagram for a very simple circuit consisting of a single \(9\text{V}\) battery connected to a \(2\Omega\) resistor. When drawing a circuit diagram (or making a real circuit), one connects the various components together (e.g. batteries and resistors) with segments of wire that have zero resistance , even if, in practice, wires always have some resistance. However, since the wires are connected in series with resistors (or other components that have a resistance), one can always include the resistance of the wires by adding it to the resistance of the other components. For example, in Figure \(\PageIndex{4}\), if the wires have a total resistance of \(1\Omega\) , we could simply model the circuit as if the resistor had a resistance of \(3\Omega\) instead of \(2\Omega\) . In practice, this is usually accounted for when a circuit diagram is made (i.e. any resistors include the resistance of the wires connected to it).
The circuit in Figure \(\PageIndex{4}\) is simple to analyze. In this case, whichever charges exit one terminal of the battery, must pass through the resistor and then enter the other terminal of the battery. We always use conventional current to analyze a circuit. Thus, we model the circuit as if positive charges exit the positive terminal of the battery, go through the resistor, and then enter the negative terminal of the battery.
We recommend that you always draw a “battery arrow” for each battery in a circuit diagram to indicate the direction in which the electric potential increases and in which direction the conventional current would exit the battery if a simple resistor were connected across the battery. In complex circuits, the current may not necessarily flow in the same direction as the battery arrow, and the battery arrow makes it easier to analyze those circuits. We also indicate the current that is flowing in any wire of the circuit by drawing an arrow in the direction of current on that wire (labeled \(I\) in Figure \(\PageIndex{4}\)).
It is helpful to think of the value of the electric potential along different parts of a circuit, as illustrated in Figure \(\PageIndex{5}\) for the same circuit as in Figure \(\PageIndex{4}\).
Since the wires have no resistance, the electric potential is constant along a wire. In other words, because the wire has no resistance, the charges/current cannot dissipate any power in the wire ( \(P=I^2R\) ), and the charges do not “loose” any potential energy (and the potential thus cannot change). The only place where the charges can dissipate energy is inside the resistor. Once the charges have crossed the resistor, the electric potential in the wire is again constant until they reach the other terminal of the battery. Thus, in this simple circuit, the electric potential difference across the resistor is the same as the potential difference across the terminals of the battery. This is shown by the colored areas in Figure \(\PageIndex{5}\). If we choose \(0\text{V}\) to be defined at the negative terminal of the battery, then the potential is \(9\text{V}\) everywhere in the red area (to the right of the resistor), and \(0\text{V}\) everywhere in the grey area (to the left of the resistor).
We can apply Ohm’s Law (the macroscopic version) to the resistor and determine the current in the circuit, since we know the potential difference across the resistor:
\[\begin{aligned} \Delta V&=RI\\[4pt] \therefore I&=\frac{\Delta V}{R}=\frac{(9\text{V})}{(2\Omega)}=4.5\text{A}\end{aligned}\]
It is helpful to think of circuits in terms of energy. Charges move along the circuit and their potential energy changes as they go through components, while it remains constant as they move through a wire. If a positive charges enters the negative terminal of a battery and exits the positive terminal, its potential energy will have increased. If that charge then enters a resistor, its potential energy will decrease as it moves through the resistor, since the charge will “use” its potential energy to heat up the resistor. Batteries provide the energy to “push” the charges through the resistors in the circuit by converting chemical potential energy into the electrical potential energy of the charges.
It is also useful to make the analogy with fluid dynamics; one can think of the battery as a pump that is continuously pushing a viscous incompressible fluid through a pipe with a narrow section, as illustrated in Figure \(\PageIndex{6}\). The wide section of the pipe is akin to the wires with no resistance, and the narrow section is akin to the resistor. The pressure difference generated by the pump is analogous to the voltage produced by the battery, and the flow rate of the liquid is analogous to the electric current. The pressure in the pipe does not drop in the wide section, if there is no resistance. The entire pressure drop of the fluid is across the narrow section, just as the voltage only drops across the resistor.
Example \(\PageIndex{1}\)
Two resistors, of \(2\Omega\) and \(4\Omega\) , respectively, are connected in series to a \(12\text{V}\) battery. What is the current through each of the resistors, and what is the voltage across each resistor?
Solution
We start by making a circuit diagram, as in Figure \(\PageIndex{7}\), showing the resistors, the current, \(I\) , the battery and the battery arrow. Note that since this is a closed circuit with only one path, the current through the battery, \(I\) , is the same as the current through the two resistors.
If we choose the potential on the negative side of the battery to be \(0\text{V}\) , then points \(a\) and \(e\) on the diagram are at a potential of \(0\text{V}\) , since potential cannot change in a wire with no resistance. Similarly, the points at \(b\) and \(c\) are at a potential of \(12\text{V}\) (relative to points \(a\) and \(e\) ). At point \(d\) , between the two resistors, the potential will be between \(0\text{V}\) and \(12\text{V}\) , since the potential will “drop” as the current goes through the \(2\Omega\) resistor.
The easiest way to determine the current through this simple circuit is to combine the two resistors into a single effective resistor with resistance:
\[\begin{aligned} R_{eff}=(2\Omega)+(4\Omega)=6\Omega\end{aligned}\]
so that the circuit can be simplified to that shown in Figure \(\PageIndex{8}\):
The potential difference across the effective resistor is the same as that across the battery (between points \(e\) and \(c\) ), so that Ohm’s Law can be applied to the effective resistor to determine the current that traverses it:
\[\begin{aligned} \Delta V &= R_{eff}I\\[4pt] \therefore I&=\frac{\Delta V}{R_{eff}}=\frac{(12\text{V})}{(6\Omega)}=2\text{A}\end{aligned}\]
This current is the same that traverses each individual resistor, since it is the same as the current that goes through the battery. Referring back to the full circuit (Figure \(\PageIndex{7}\)), we can now use Ohm’s Law to calculate the voltage drop across each resistor, since we know the current through each resistor. The voltage across the \(2\Omega\) resistor is given by:
\[\begin{aligned} \Delta V_{2\Omega}=RI=(2\Omega)(2\text{A})=4\text{V}\end{aligned}\]
and the voltage across the \(4\Omega\) resistor is given by:
\[\begin{aligned} \Delta V_{4\Omega}=RI=(4\Omega)(2\text{A})=8\text{V}\end{aligned}\]
Note that the sum of these two voltages is equal to the voltage increase across the battery, by conservation of energy. Consider the electric potential at different points in Figure \(\PageIndex{7}\) as you move clockwise around the loop starting at point \(a\) . If the electric potential is defined to be \(0\text{V}\) at the negative end of the battery (points \(a\) and \(e\) ), the potential at point \(d\) (between the resistors) is the potential at point \(e\) plus the potential difference across the \(4\Omega\) resistor:
\[\begin{aligned} V_d = V_e+\Delta V_{4\Omega}=(0\text{V})+(\Delta V_{4\Omega})=8\text{V}\end{aligned}\]
If we then add the potential difference across the \(2\Omega\) resistor to the potential at point \(d\) , we find that the potential at point \(c\) is \(V_c=V_d+\Delta V_{2\Omega}=12\text{V}\) , as expected, since this corresponds to the potential at the positive terminal of the battery.
Discussion
In this example, we showed how one can model a circuit by combining resistors together into effective resistors to simplify the circuit. We also showed how the potential differences across different components in a circuit must add up to zero (the voltage drops across the resistors must sum to the voltage increase across the battery).
Exercise \(\PageIndex{1}\)
What is the voltage across the combination of a \(3\text{ V}\) battery connected in series with a \(6\text{ V}\) battery, where the negative terminal of the \(6\text{ V}\) battery faces the positive terminal of the \(3\text{ V}\) battery?
- \(9\text{ V}\).
- \(6\text{ V}\).
- \(3\text{ V}\).
- \(0\text{ V}\).
- Answer
The real battery in a circuit
So far, we have modeled batteries as “ideal” devices that provide a fixed potential difference. In reality, this neglects the fact that the materials that make the battery will themselves have a resistance. For example, if electrons want to leave the zinc rod in the electric cell illustrated in Figure \(\PageIndex{1}\), they will loose some energy as they pass through the zinc. Thus, when modeling a real battery in a circuit, it is important to include its “internal resistance”, as a resistor in series with the potential difference. This is illustrated in Figure \(\PageIndex{9}\), which shows the two terminals of a real battery, an ideal battery (with a fixed potential difference, \(\Delta V_{ideal}\) ), and its internal resistance, \(r\) (which can be drawn on either side of the battery).
It is important to note that the potential difference across the terminals of the real battery is only equal to the potential difference across the ideal battery if there is no current flowing through the battery . If there is a current, \(I\) , flowing through the internal resistance, the electric potential will decrease by an amount \(Ir\) across the internal resistance, and the voltage across the real terminals will be \(\Delta V_{ideal}-Ir\) .
Example \(\PageIndex{2}\)
When no resistance is connected across a real battery, the potential difference across its terminals is measured to be \(6\text{V}\) . When a \(R=2\Omega\) resistor is connected across the battery, a current of \(2\text{A}\) is measured through the resistor. What is the internal resistance, \(r\) , of the battery, and what is the voltage across its terminals when the \(R=2\Omega\) resistor is connected?
Solution
The real battery can be modeled as an ideal battery with potential difference, \(\Delta V_{ideal}\) , in series with an internal resistance, \(r\) . While we do not know the value of the internal resistance, we are told that the potential difference across the terminals of the real battery is \(6\text{V}\) when no current flows through it . Since no current flows through the internal resistance, the voltage does not drop across the internal resistance, and the voltage across the terminals of the real battery (e.g. Figure \(\PageIndex{9}\)) must thus be equal to the voltage across the terminals of the ideal battery, so that \(\Delta V_{ideal}=6\text{V}\) .
With this information, we can make a circuit diagram for the case when the \(2\Omega\) resistor is connected across the terminals of the real battery, as in Figure \(\PageIndex{10}\).
The terminals of the real battery are located at points \(a\) and \(c\) of the diagram, whereas the terminals of the ideal battery corresponds to points \(a\) and \(b\) . When no current flows through the internal resistor, \(r\) , there is no voltage drop across that resistor and the potential at \(b\) will be equal to the potential at \(c\) , as we argued above.
The circuit in Figure \(\PageIndex{10}\) is now identical to that analyzed in Example 20.1.1 , and can be treated the same way. We can combine the \(2\Omega\) resistor with the internal resistance, \(r\) , in series to obtain an effective resistor, \(R_{eff}=r+R\) . The voltage drop across the effective resistor will be the same as the potential difference across the ideal battery, and we can make use of Ohm’s Law to find the internal resistance, \(r\) :
\[\begin{aligned} \Delta V_{ideal}&= R_{eff}I=(r+R)I\\[4pt] \therefore r &= \frac{\Delta V_{ideal}}{I}-R=\frac{(6\text{V})}{(2\text{A})}-(2\Omega)=1\Omega\end{aligned}\]
Now that we know the internal resistance, we can determine the voltage drop across the internal resistor, using Ohm’s Law:
\[\begin{aligned} \Delta V_r = rI=(1\Omega)(2\text{A})=2\text{V}\end{aligned}\]
The voltage drop across the real terminals of the battery (between points \(a\) and \(c\) ), is thus given by:
\[\begin{aligned} \Delta V_{real}=\Delta V_{ideal}-\Delta V_r=(6\text{V})-(2\text{V})=4\text{V}\end{aligned}\]
Again, you can verify that the voltage drops across the two resistors will sum to the total voltage drop across the terminals of the ideal battery.
Discussion
Modeling real batteries is not so different from modeling ideal batteries, since one only needs to include an internal resistance into the circuit. The key difference with a real battery is that the voltage across its real terminals depends on what is connected to the battery. In the example above, the battery has a voltage of \(6\text{V}\) across its (real) terminals when nothing is connected, but the voltage drops to \(4\text{V}\) when a \(2\Omega\) resistor is connected.
Exercise \(\PageIndex{2}\)
Suppose that you would like to measure the ideal voltage of a real battery by connecting a measurement device (a voltmeter) across its terminals. In order to get the most accurate reading, should you choose a voltmeter with a high resistance, or a voltmeter with a low resistance?
- High resistance.
- Low resistance.
- It doesn’t matter if the voltmeter has a high or low resistance.
- Answer


3.1.5: Unary Representation of Numbers
Turing machines work on sequences of symbols written on their tape. Depending on the alphabet a Turing machine uses, these sequences of symbols can represent various inputs and outputs. Of particular interest, of course, are Turing machines which compute arithmetical functions, i.e., functions of natural numbers. A simple way to represent positive integers is by coding them as sequences of a single symbol \(\TMstroke\) . If \(n \in \Nat\) , let \(\TMstroke^n\) be the empty sequence if \(n = 0\) , and otherwise the sequence consisting of exactly \(n\) \(\TMstroke\) ’s.
A Turing machine \(M\) computes the function \(f\colon \Nat^n \to \Nat\) iff \(M\) halts on input \[\TMstroke^{k_1} \TMblank \TMstroke^{k_2} \TMblank \dots \TMblank \TMstroke^{k_n}\nonumber\] with output \(\TMstroke^{f(k_1, \dots, k_n)}\) .
Addition: Build a machine that, when given an input of two non-empty strings of \(\TMstroke\) ’s of length \(n\) and \(m\) , computes the function \(f(n,m) = n + m\) .
We want to come up with a machine that starts with two blocks of strokes on the tape and halts with one block of strokes. We first need a method to carry out. The input strokes are separated by a blank, so one method would be to write a stroke on the square containing the blank, and erase the first (or last) stroke. This would result in a block of \(n + m\) \(\TMstroke\) ’s. Alternatively, we could proceed in a similar way to the doubler machine, by erasing a stroke from the first block, and adding one to the second block of strokes until the first block has been removed completely. We will proceed with the former example.
Trace through the configurations of the machine for input \(\tuple{3,5}\) .
Subtraction: Design a Turing machine that when given an input of two non-empty strings of strokes of length \(n\) and \(m\) , where \(n > m\) , computes the function \(f(n,m) = n - m\) .
Equality: Design a Turing machine to compute the following function: \[\fn{equality}(x,y) = \begin{cases} \text{1} & \text{if $x = y$} \\ \text{0} & \text{if $x \neq y$} \end{cases}\nonumber\] where \(x\) and \(y\) are integers greater than \(0\) .
Design a Turing machine to compute the function \(\min(x,y)\) where \(x\) and \(y\) are positive integers represented on the tape by strings of \(\TMstroke\) ’s separated by a \(\TMblank\) . You may use additional symbols in the alphabet of the machine.
The function \(\min\) selects the smallest value from its arguments, so \(\min(3,5)=3\) , \(\min(20,16)=16\) , and \(\min(4,4)=4\) , and so on.


1: LibreTexts Fundamentals
The goal of the LibreTexts project is to establish a vetted, comprehensive, and widely-used platform for the dissemination and implementation of customized course materials for post-secondary courses, while significantly reducing existing textbook costs for students. The LibreTexts project consists of 13 independently operating and interconnected “libraries” that focus on augmenting education in separate fields via a series of open-access online textbook environments that includes complementary ancillaries that advance diverse pedagogical methods and support multiple student learning styles. The LibreTexts libraries currently benefits over 100 million students per year and is arguably the most visited chemistry website in the world.
-
- 1.3: Orientation
- The page provides a guide on navigating and utilizing the features of the LibreTexts Commons, which includes access to various libraries, tools for readability, and resources for study. It explains how to navigate through the libraries, use the blue navigation bar, adjust text settings, and utilize resources. Additionally, it details the functions of the light blue navigation bar in the book view and explains tools for creating, saving, and remixing content for educators and readers.