Unit 2: Electrostatics
Now, let's turn to the study of electricity and magnetism, two different aspects of electromagnetism. We start by looking at electrostatics: the rules that govern the behavior of static charges. Thales of Miletus (c. 624–548 bc), the Greek mathematician, astronomer, and philosopher, carried out the first experiments on electrical phenomena when he observed that you can generate a static charge when you rub amber with wool.
Completing this unit should take you approximately 20 hours.
Upon successful completion of this unit, you will be able to:
- state Coulomb's Law and the units of the physical quantities contained in the law;
- solve problems involving electric forces, electric fields, and electric potentials;
- compare and contrast the electric potential and the electric field;
- solve problems involving the motion of charged particles in an electric field;
- define capacitance and the factors that determine capacitance;
- describe the effect of a dielectric material in a capacitor; and
- define electric potential energy and how capacitors can be used to store energy.
2.1: Introduction to Electricity
As you know, the atoms that make up physical objects include protons, electrons, and neutrons. Protons are positively charged, electrons are negatively charged, and neutrons are neutral. Consequently, all things are made up of charges.
Opposite charges attract one another (negative to positive). Like charges repel one another (positive to positive or negative to negative). Most of the time, positive and negative charges are balanced in an object, making that object neutral.
Static electricity results from an imbalance between the negative and positive charges in an object. These charges can build up on the surface of an object until they find a way to be released or discharged. One way to discharge them is through a circuit.
Static electricity refers to a constant imbalance of electric charges within or on the surface of a material. The charge remains until it can move away, such as via an electric current or electrical discharge.
Rubbing certain materials against one another can transfer negative charges (electrons). For example, when you rub your shoe on the carpet, your body collects extra electrons. The electrons cling to your body until they can be released. When you reach out and touch an animal, you get a shock as you release the surplus electrons to your unsuspecting pet. An accumulation and discharge of static electricity can create a spark that can be dangerous, especially when a flammable fluid like gasoline is around. Lightning is an example of static electricity. Read this text, which explores static electricity and charge.
The reading in our textbook explained what a battery is at a basic level. The important thing to keep in mind is that batteries can be charged and discharged, but they do not create new charges. Batteries contain a mechanism (usually chemical) that can maintain a charge imbalance between the positive and negative terminals even if you allow charges to flow between the two terminals through an external circuit like a light bulb. When a battery "loses its charge", that means that the chemical process that was designed to maintain the charge imbalance has stopped working.
Charge is a property of fundamental particles found in all the matter around us, and in that sense, charge is not so different from mass. The big difference between the mass and charge of a particle is that there are no negative masses. On the other hand, whereas the protons in the nucleus of an atom carry positive charge, the electrons that inhabit the outer regions of an atom are negatively charged.
To create a charge on a macroscopic object such as a balloon, we just transfer electrons from one object to another while leaving the positively-charged atomic nuclei where they are. When this is achieved by friction, it is called triboelectricity.
Because charge is ultimately carried by fundamental particles, it also obeys a fundamental conservation law: charge cannot be created or destroyed. Watch this video, which gives several examples of conservation of charge.
2.2: Conductors and Insulators
A conductor is a material that allows electrons to flow freely through them from particle to particle. The moving electrons may lose some energy when they collide with the fixed atoms and molecules, but they can move in a conductor. Superconductors allow the movement of charge without any loss of energy. For example, electric current can flow freely through a conductor. Examples of conductors include metals such as aluminum, copper, gold, iron, silver, and steel. Salty water and molten salt are also conductors.
Insulators, in contrast, are made from materials that lack conduction electrons. The electrons and ions in insulators are bound in the structure and cannot move easily. The charge moves with great difficulty, if at all. Examples of insulators include most non-metallic solids, such as amber, fur, glass, plastic, porcelain, rubber, wood, and most semi-precious gems. Pure water and dry table salt are also good insulators.
Neutral objects are not "devoid" of charge – they just contain balanced amounts of positive and negative charges. Watch this video for more illustrations.
2.3: Coulomb's Law
We can detect the presence of an electric charge by the forces they exert on other charged objects. We call these forces electric forces. These forces depend on how far away you are, just like the gravitational force between two planets depends on their separation. Like the law of universal gravitation that Newton used to explain how planets move around the Sun, there is also a force law of electricity that explains how electrons are held by the nucleus inside an atom (even though that requires additional ingredients from the modern theory of quantum physics).
The strength of the electric force between two charged spheres depends on their separation in the same way as the gravitational force between two spheres: an inverse-square law. This means that if you double the distance, the force decreases by a factor of four. The mathematical formula for electrostatic force is named after the French physicist Charles Coulomb (1736–1806), who performed experiments and first proposed a formula to calculate it.
The analogies between gravitational and electrostatic force go even further, in that the force between two objects is proportional to the charge of each object individually. So when you double one of the charges, the force doubles. If you double both charges, the force between them quadruples.
The most obvious difference between gravity and electricity is that gravity is always attractive because mass does not come with two different signs. According to Coulomb's Law, the electric force is always attractive between charges of opposite sign and always repulsive between charges of the same sign!
We can discover a less obvious difference between gravity and electricity if we try to quantify the forces in units of Newtons. It then turns out that the force of gravity is unimaginably weaker than the electrostatic force at the atomic level. Read this text, which gives details.
In the quantitative discussion, we encountered the standard unit of charge, the Coulomb, named after the person who discovered the electric force law. Watch this video, which drives home the point that a Coulomb is actually a huge amount of charge if you count the number of fundamental microscopic particles that would have to be gathered to make that amount of charge.
Coulomb's force law contains the assumption that the two charged objects involved in the interaction are either spherical (charged uniformly) or are so far apart that it is okay to consider both of them as point-like (that is, objects without a shape altogether). Then the direction of the electric force is always along the straight line connecting the objects. In the case of spheres, we consider that to be from center to center.
It can be confusing to figure out in what direction the force should point. To figure out the correct force direction, remember this simple rule: the forces at opposite ends of the straight line connecting the charges must always point in opposite directions. That is the same as for the tension forces at both ends of a spring. Either both ends are pulling inward, or both ends are pushing outward. Since two charges with opposite signs attract, the forces in Coulomb's Law point inward at both ends of the line connecting the charges. And since two charges with equal sign repel, the Coulomb force, in that case, points outward at both ends of the connecting line.
You may ask yourself how Coulomb's Law can be true for the electrons and protons inside a material if it is electrically neutral overall. After all, even though electric forces fall off with increasing distance, the force is never precisely zero, no matter how far away you are from an individual point-like charge.
In fact, when we say an object is electrically neutral, it is really a statement about the electrical forces that the object is (or is not) able to create over relatively large distances. To measure such a force, you would approach the neutral object with an electrically charged test object and observe if it feels a push or a pull. When you are far enough away from a neutral object, the repulsive and attractive forces created by the positively- and negatively-charged fundamental particles inside of it cancel out when their combined effect on your test particle is measured. The combined effect of several simultaneous forces is what we call the resultant, and the resultant of two opposing forces of equal strength is zero.
Almost all applications of electrostatics (including the explanation of what "neutral" means) involve large numbers of charges. In contrast, Coulomb's Law only applies to the limited special scenario of two charged objects that essentially behave like points. This is why we need additional tools to make practical use of Coulomb's Law.
2.4: Electric Field and Gauss' Law
When Newton chronicled the law of universal gravitation, he also explained the well-known observation that all objects fall at the same acceleration on Earth's surface. Without going into the details, the crucial fact in the explanation was that the gravitational force on any test object is proportional to the mass of the test object.
Therefore, when the gravitational force is divided by the mass of the test object, you get a quantity that is independent of the test object itself. For gravity, that is just the universal value of acceleration. It only depends on the properties of the other object involved in the gravitational pull: the Earth.
Now, according to Coulomb's Law, the electrostatic force on a test object due to some other object is proportional not to the mass but to the charge of the test object. Following the same logic as for gravity, what do you get if you divide the Coulomb force by the charge of the test object? The result is again a quantity that is independent of the test object itself. But this time, it is not the acceleration. It is called the electric field (or E-field), and it is wholly determined by the charge of the other object involved in the electrical interaction.
As we mentioned earlier, electric fields are a tool that helps us deal with situations where Coulomb's Law has to be applied to collections of several charged particles at once. Read this section, which discusses this concept and introduces a powerful way of visualizing the effects of charges on other charges: electric field lines. There is a new level of abstraction here because although the lines representing the electrostatic field begin and end on charged objects (more or less tangible entities), the lines themselves fill the empty space between the charges. So, we are drawing something that is not tangible because there appears to be nothing there.
To put it briefly: electric fields are would-be electric forces. As a physical quantity, the electric field at any given point in space tells you what the electric force on a test object would be if you were to place it there.
Based on how the electric field is constructed (remember the analogy to Newtonian gravity), you get the electric force on an object of charge Q in an electric field of strength E by simply multiplying the two. The purpose of the field lines is to tell you in which direction that electric force would then point.
The use of electric field lines goes beyond mere visualization. Drawing field lines for a collection of electric charges can help you identify patterns and symmetries that make calculations easier in practical applications.
As an example for symmetry in electric field line patterns, recall that the field lines of a point charge form a star shape that looks the same in all directions. The same field line pattern also forms around a uniformly charged sphere, which is why Coulomb's Law makes no distinction between such a sphere and a point charge. It doesn't matter whether the sphere has all the charge sitting on its surface or if the charge is spread out throughout its interior.
When we look at the symmetries of field line patterns, we discover that you can get the same electric fields from seemingly quite different distributions of charges.
2.5. Applications of Electrostatics
Electrostatics (also known as static electricity) is the branch of physics that deals with stationary electric charges. This means it involves charges whose distribution in space stays constant over time. Although Coulomb's Law is the foundation of electrostatics, it is not always easy to apply when large numbers of charged particles are involved.
Using field lines as a tool, we can draw conclusions about an electric field even when the charges are crowded so densely that they lose their individual identity. This happens when charges are transferred to a conductor. We approach this situation from what we know about the electric field itself, rather than from the individual charges, as the cause of the electric field. We can do this with the help of some additional assumptions. The main assumption is that in an electrical conductor in electrostatic equilibrium, charges will arrange themselves on the surface so that their resultant electric field lines point exactly perpendicular to the surface.
Read this text, which discusses some consequences of this approach.
Watch this lecture to review this material and connect it to applications of electrostatics.
Read this section, which explains (among other applications) how inkjet printers work.
2.6: Electric Potential and Electric Potential Energy
Electric potential energy refers to the energy something has due to its position in an electric field (the capacity for doing work due to its position or configuration). For example, if you have a positive charge (+ charge) and you move it near another positive charge, it will want to deflect. If you push it closer to the positive charge, it will want to deflect more. When the charged object is let go from rest, it will speed up and gain kinetic energy by an amount equal to the change in its electrical potential energy. As for all forms of energy, the standard unit of electrical potential energy is the Joule.
Any test object can have an electric potential energy only if it feels an electric force. Recall that the electric force on a test object is proportional to its charge. This makes the electric potential energy proportional to the charge, too! Therefore, one can play the same game with the potential energy that led us from the electric force to the electric field: divide by the charge to get a quantity that is independent of the test object.
The result is the electric potential. It is to the electric potential energy what the electric field is to the electric force. In other words, electric potential is the would-be electric potential energy. Similar to the electric field, the electric potential depends on the distribution of all the charges that are present before we decide to introduce an additional test charge to probe what those existing charges would do to it.
As with any form of potential energy, you have to choose a reference point to specify its value. The choice of reference point is completely arbitrary because only changes in electrical potential have any physical significance. The change in electric potential between two points is also called the voltage between those points.
Read this text, which discussed these concepts.
Watch this lecture, which gives a visual summary of how electrostatic potential energy can be converted to kinetic energy.
The video we just watched on electrostatic potential also makes use of an important property that electrical conductors show as relates to the electric potential: in equilibrium, the electric potential is the same everywhere along the surface of a conductor. This is closely connected to the observation that electric field lines are perpendicular to the conducting surface. Both of these facts come together in the discussion of the parallel-plate configuration in the next section.
Watch this lecture, which leads us from the parallel-plate configuration back to the point-like charges that we started with when introducing Coulomb's Law. The reason we discussed the potential for the seemingly more complicated situation of densely crowded charges on parallel plates before going back to the individual point charge is math. We found that the potential between two plates changes in direct proportion to the distance from one of the plates. This is connected to the fact that the electric field lines between the plates are parallel straight lines.
For a point charge, recall that the field lines form a radial star shape. As a consequence, the potential in this situation is more difficult to compute. Fortunately, the result is relatively simple, differing only slightly from Coulomb's formula for the electric force of a point charge. The main difference is that the potential decreases with the inverse distance to the point charge, not with the inverse squared distance.
Read this text, which illustrates an example of this concept and gives a written formula.
The second part of the video we just watched addresses the concept of equipotential lines, which you can find in this next section of the text.
As you read, pay attention to the interactive simulation at the bottom of the page. Try recreating the charge configurations of Figures 19.9 and 19.10 by yourself. Then click on the panel on the right to enable the "Voltage" indicator and drag the indicator around the screen. To find an equipotential line, try to move the voltage indicator in just the right way so that its reading stays roughly the same. As you do, note what shape you are tracing out. It ought to be similar to the closed green lines in the figures, indicating the equipotential lines.
2.7: Capacitors and Capacitance – Storage of Electric Energy
A capacitor (also called a condenser) is a device that stores electric charge in an electric field. It is a passive electronic component with two terminals. Although they work in different ways, a capacitor looks like a battery, which also stores electrical energy. Inside the capacitor, the terminals connect to two metal plates separated by a non-conducting substance, or dielectric.
The dielectric dictates what kind of capacitor it is and for what it is best suited. Depending on the size and type of dielectric, some capacitors are better for high frequency uses; others are better for high voltage applications.
Watch this video, which introduces the central characteristics of capacitors.
Read this text, which gives examples of capacitors. Air capacitors power radio tuning circuits; mylar capacitors power timer circuits, such as clocks, alarms, and counters; glass capacitors power high voltage applications; ceramic capacitors are used for high-frequency antennas X-rays, and MRI machines; super-capacitors power electric and hybrid cars.
Capacitance is the amount of charge stored per unit volt. This physical quantity characterizes how capacitors perform in electric circuits, which we will discuss in more detail later.
When viewed as a whole, is a capacitor that stores a certain amount of charge electrically neutral or electrically charged?
The answer is that the two plates of a capacitor store the same amount of charge but with opposite signs. That means that the capacitor is actually neutral, even when it is said to store a large amount of charge! If you could probe the space between the capacitor plates, you would feel electric forces because each plate individually is charged.
But viewed from the outside (at a distance much larger than the plate separation), the forces from the oppositely charged plates would cancel each other, so there is practically no electric field. This is why capacitance, and not field strength, is the quantity that we care most about when using capacitors in practice.
It is not usually feasible to build capacitors just by placing two parallel metal plates opposite each other with an air gap in between. Instead, you fill the space between the conductors with a different insulating material, called a dielectric. This is not just for mechanical stability – it actually enhances the functionality of the capacitor.
Watch this lecture to see how this works.
Although you will typically see capacitors illustrated as two parallel plates arranged like a sandwich, many realistic capacitor designs look nothing like that. All capacitors have two conducting sheets and two-wire terminals that connect to those sheets. But the internal geometry of the sheets can be quite intricate – for example, arranged like stacked cups or rolled-up tape. After this introduction to capacitors, watch this lecture, which closely follows the presentation in the text.
This section gives formulas that combine several capacitors when connected in different ways. The idea is that, from the application point of view, all that matters about a capacitor is the value of its capacitance (as mentioned earlier, the shape of the conductors inside a capacitor can vary widely). That makes it possible to characterize entire collections of capacitors by a single capacitance as if they were just a single unit with two terminals that connect it to the rest of the world.
The two basic ways of combining capacitors are in series and in parallel. Watch this video to see an example calculation for capacitors in series.
The ability to store charge in a capacitor is only one aspect of its usefulness. The other aspect is its ability to store energy. The two aspects are related because it takes work to charge a capacitor. Charging really means that we create a charge imbalance between the plates by moving excess electrons (negative charges) onto one plate against their mutual repulsion, and simultaneously removing electrons from the other plate against the attractive force of the positively charged atomic nuclei that remain there.
Working against these forces is analogous to lifting a weight against gravity – except that gravity stays constant, whereas electric forces change depending on the amount of charge that has already been moved.
Read this text, which explores the relationship between stored charge and stored energy in a capacitor.
Watch this video for two numerical examples (you may skip the second example at the end).