Lecture notes · Physics: Electricity & Magnetism · Lecture 6

Capacitors

Yale PHYS 201 Fundamentals of Physics II, lecture 6, explained in plain English

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Part 1 · 0:00–10:00

This part explains how electric potential works as an energy idea, and how it links to the electric field, with a correction to an argument from the last lecture.

Key ideas

A conservative force is one where the work done around any closed loop adds up to zero, so you can define a potential. The electric field has this property, so a potential can be defined for it. 0:49
The energy that matters is the charge times the potential, not the potential alone. The potential is like an electrical height, and the charge is like a mass. 1:31
The minus sign in the formula means the work you do when you push a charge against the field. The field pushes one way, and you must push the opposite way to move the charge. 3:31
For a single point charge, the potential depends only on distance: it is the charge divided by 4π times ε0 and by the distance r. For many charges, you add up each one's contribution. This adding is called superposition. 5:08
You can go from potential to electric field by taking a derivative, called the gradient. You can go from field to potential by doing an integral. 6:29

Words to know

potential: An energy-like value at each point in space. It is the energy per unit charge, so it tells you how much work it takes to move a charge to that point.
conservative: A force is conservative when the total work it does around any closed loop is zero. This means the energy can be stored and given back, like a ball rolling down and back up a hill.
gradient: A derivative that works in more than one direction. It gives a vector that points the way a value changes fastest, and is used here to get the electric field from the potential.
superposition: The rule that the total effect of many sources is found by adding up the effect of each source separately.

Example

Think of potential as the height of a hill. A heavier ball and a lighter ball at the same spot have the same height, but the heavier one stores more energy, so the energy depends on both the height and the mass. In the same way, the potential is a height, and the energy depends on the charge you carry.

3 checked practice questions for this part are in the app.

Part 2 · 10:00–20:00

This part explains why the work done by an electric field stays the same no matter which path you take, and why electric potential is an easier tool than electric field.

Key ideas

Many tiny differences can add up to a real difference. A zigzag path is longer than a straight one, even though each small step looks almost the same. 10:00
You can change the path you were given into a different path, as long as the work done stays the same. Path length can change while the work does not. 10:29
Moving at a slant to the field only counts the part of the move that goes along the field. So the work depends only on how far you move radially, not on the exact route. 12:55
Units such as newtons per coulomb for electric field and volts (joules per coulomb) for electric potential help you check your work. Always write units. 13:39
Conservation of energy lets you skip the steps in between. You only need the starting and ending energies to find an unknown, such as a final speed. Adding potentials is also easier than adding field arrows, and you then get the field by taking derivatives. 17:03

Words to know

dot product: A way to multiply two arrows (vectors) that gives one number. It depends on how much the two arrows point in the same direction.
electric potential: The work needed to move a unit charge from one place to another. It is a single number at each point, so it is easier to add up than electric field arrows.
volt: The unit of electric potential. One volt means one joule of work per coulomb of charge.
derivative: A way to find how fast a quantity changes as you move in one direction. The lecture uses it to get the electric field from the electric potential.

Example

Imagine walking up a hill. Whether you take a winding trail or a steep straight climb, the height you gain is the same, even though the paths are different lengths. The lecture says work works the same way: the path can change, but the result stays the same.

3 checked practice questions for this part are in the app.

Part 3 · 20:00–30:00

This part shows how working with electric potential, a single number at each point, often makes electric fields easier to find than working with the fields directly.

Key ideas

Far away, a pair of opposite charges (a dipole) makes a potential that falls off like 1 over r squared, and it depends on the angle theta. The lesson shows this by subtracting two distances that are almost equal. 22:27
Once you have the potential as a function of position, you get the electric field by taking derivatives. The lesson says derivatives are easy to take, while integrals are hard. 23:09
For a ring of charge, you add up the small potentials from each piece. Potential is a plain number, so you add numbers, not arrows. 24:18
For a hollow charged sphere, it is easier to find the electric field first, using Gauss's Law, and then get the potential from it. Inside the sphere the field is zero, so the potential stays at one fixed value. 26:35
Between two oppositely charged parallel plates, the two fields point the same way and add together. The lesson uses this picture to show how a field can be visualized. 29:58

Words to know

potential: A single number at each point in space that tells you how much energy a charge would have there. It is a scalar, so it has size but no direction.
electric field: The push a charge would feel at each point in space. It has both a size and a direction, so it is a vector.
Gauss's Law: A rule that links the electric field through a surface to the charge inside that surface. The lesson uses it to find the field from symmetry.
dipole moment: A number that measures how strongly a pair of opposite charges is separated, written as p in the lesson.

Example

Think of a hilly park. The height at each spot is like the potential, one number per spot. To know which way a ball would roll, you look at how steeply the ground slopes, which is like taking a derivative. It is easier to draw the height map first and then work out the slopes than to draw every slope arrow directly.

3 checked practice questions for this part are in the app.

Part 4 · 30:00–40:00

This part shows how electric fields and voltage (potential) relate, using the picture of climbing a hill and comparing it to gravity.

Key ideas

Between two charged plates, the electric field is trapped between them and is twice as strong as it would be next to a single sheet. Outside the plates, the fields cancel out. 30:00
Voltage is the same everywhere across a flat plane that is perpendicular to the field. You can move sideways on that plane without doing any work. So the lines of equal voltage run horizontally while the field lines run vertically. 30:53
An electric field works like gravity. Voltage rises as you go up, and a charge that falls through a voltage gains energy. For example, one coulomb dropping through 4 volts picks up 4 joules. 33:13
An electron has a negative charge, so the field pushes it the opposite way from a positive charge. A released electron moves uphill toward higher voltage and loses potential energy as it goes. 34:27
In general, lines of equal voltage are always perpendicular to the electric field lines. This is true even when the setup is not symmetrical. 37:57

Words to know

potential: The voltage at a point. It is the work needed to bring a unit of charge from very far away to that point.
coulomb: The standard unit for an amount of electric charge. It is a large amount, made of many tiny electrons.
electron volt: A very small unit of energy. It is the energy an electron gains when it falls through 1 volt. It is handy for measuring tiny charges.
equipotential: A surface or line where the voltage is the same at every point. You can move along it without doing any work.

Example

Think of a ball rolling down a hill. Gravity pulls it downhill, and the higher you start, the more energy it gains. Voltage works the same way: a charge falls down a voltage hill and gains energy along the way.

3 checked practice questions for this part are in the app.

Part 5 · 40:00–50:00

This part explains how electric potential works like a map of hills and valleys, and what happens to electric charges inside metal (conductors).

Key ideas

The electric field points in the direction where potential drops fastest. Think of it like water running straight downhill. 40:01
Moving sideways, perpendicular to the field, keeps you at the same potential. You do no extra work doing this, so the potential does not change. 41:33
In a solid metal, free charges keep moving until they cancel out any electric field inside. Once the field is zero, they stop moving. 44:27
A solid conductor has no charge inside it. Any charge sits on its surface, and a hollow space inside has no field. 45:44
The whole metal sits at one single potential. If it did not, there would be an electric field inside it, and that is not allowed. 48:43

Words to know

gradient: A arrow that points in the direction where something increases the fastest. It tells you the steepest way up a hill.
equipotential: A line or surface where the potential (electric height) is the same everywhere. It is like a contour line on a map.
conductor: A material, like metal, where electrons are free to move around inside it.
line integral: A way of adding up a value along a path. Here it is used to check whether the path you take changes the answer.

Example

Imagine a hiker on a mountain. Water runs straight downhill along the steepest path, which is like the electric field pointing toward lower potential. If the hiker walks along a line that stays at the same height, like a contour line on a map, they do not go up or down at all.

3 checked practice questions for this part are in the app.

Part 6 · 50:00–1:00:00

This part shows a trick for solving problems with a charge near a metal sheet, and how much work it takes to gather charges together.

Key ideas

Electric field lines cannot pass into metal. They must end on the metal surface, which pulls negative charge up from the ground. 50:26
A dipole (a positive and a negative charge side by side) has a middle line where the voltage is zero. Since a metal sheet can sit on a zero-voltage line without changing anything, the dipole's field can solve the metal-sheet problem. 51:36
The force between the charge and the metal sheet is the same as the force from a hidden mirror-image charge of opposite sign. The image charge is the same as the first one, but negative. 53:40
Bringing charges together from far away takes work. Each new charge has to push against the fields of the charges already placed, and each pair of charges is counted once in the total. 55:21
To build up a charged sphere, each small bit of new charge must be pushed against the voltage of the sphere's surface. The work for each small bit is that voltage times the bit's size. 59:10

Words to know

equipotential: A surface where the voltage is the same everywhere on it. No force pushes charges along it.
conductor: A material, like metal, that lets electric charge move freely through it.
dipole: Two equal charges of opposite sign placed a small distance apart.
image charge: A made-up charge placed behind a surface to copy the effect of the real surface, like a reflection in a mirror.

Example

Think of a mirror. You see a reflection that looks like a second you standing behind the glass, and the mirror is what makes it look real. In the lesson, the metal sheet works the same way: the charge acts as if a made-up partner sits behind the sheet, and the force on the real charge is the same as if that partner were really there.

3 checked practice questions for this part are in the app.

Part 7 · 1:00:00–1:12:13

This part explains how a capacitor stores charge and energy, how to measure its capacitance, and how to calculate the energy it holds.

Key ideas

A capacitor is a way to store charge and energy, a bit like water stored high in a tank. You do work to put energy in, and you can get it back later. 1:01:38
Capacitance is the ability of a system to hold charge. You find it by dividing the charge moved by the voltage difference it creates, C = Q/V. It is measured in farads. 1:04:12
For a parallel plate capacitor, the capacitance is the permittivity constant times the plate area divided by the gap between the plates, C = ε0A/d. You find it by moving charge from one plate to the other, finding the electric field between them, and multiplying by the gap to get the voltage. 1:06:09
For two concentric spheres, the same steps give C = 4πε0ab/(b − a). When the spheres are very large and close together, this matches the parallel plate formula, which is a useful check. 1:09:01
The energy stored in a capacitor is ½q²/C, or equivalently ½CV². Each new bit of charge is harder to add than the last, so the total work is only half of what you would expect if the first charge's effort were used for every bit. 1:11:52

Words to know

capacitor: A device that stores electric charge and energy by holding opposite charges on two separate conductors.
capacitance: How much charge a system can hold for each volt of voltage difference between its parts.
farad: The unit of capacitance. It is a very large unit, so labs usually use microfarads.
potential difference: The voltage between two points. It is the work needed to move a charge from one point to the other, divided by that charge.

Example

Think of a water tower. You pump water up to the top, which takes work, and that water now holds energy. When you open the tap, the water falls and turns a turbine that gives the energy back. A capacitor works the same way: you push charge onto it with work, and it gives that energy back later.

3 checked practice questions for this part are in the app.

← Lecture 5: The Electric Potential and Conservation of Energy · Lecture 10: Ampere's Law →

Notes written by AI (Claude) from the lecture's official transcript, so check anything important against the lecture. Open Yale Courses · CC BY-NC-SA 3.0 · not affiliated with or endorsed by Yale. These notes are shared under the same license.

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