Lecture notes · Physics: Electricity & Magnetism · Lecture 5

The Electric Potential and Conservation of Energy

Yale PHYS 201 Fundamentals of Physics II, lecture 5, explained in plain English

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Part 1 · 0:00–10:00

This part explains how to predict how a charge moves in an electric field, and how energy conservation gives a shortcut for finding its speed.

Key ideas

If you know the electric field everywhere, you can find the force on a moving charge. Multiply the charge by the field at its location, and Newton's law then tells you how it accelerates. 0:43
In a spring problem, you can skip the step-by-step tracking of motion. Energy conservation gives the speed at a new position directly. 5:16
The work-energy theorem says that the change in kinetic energy (the energy of motion) equals the integral of the force over the distance the body moves. 7:06
If you rearrange the work-energy result, the combination of kinetic energy and a function G stays the same as the particle moves. G is any function whose derivative is the force. 9:09
Defining potential energy U as minus G gives the energy conservation rule K + U stays constant. The force is minus the derivative of potential energy. 9:59

Words to know

electric field: A map of the push or pull that a charge would feel at each point in space. You find it by adding up the effects of all the charges nearby.
kinetic energy: The energy an object has because it is moving. It depends on the object's mass and speed.
potential energy: Stored energy that depends on where an object is, such as a stretched spring or a charge sitting in a field. The lesson calls it U.
work energy theorem: A rule saying that when a force moves a body, the change in its kinetic energy equals the total effect of that force along the path.

Example

Think of a ball rolling down a hill, which is an analogy and not part of the lesson. Its height tells you its stored energy, and its speed tells you its moving energy. When the ball rolls down, stored energy turns into moving energy, but the total stays the same.

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Part 2 · 10:00–20:00

This part explains how energy can be tracked with a conservation law, why friction does not fit that pattern, and why the energy idea gets harder to use in two dimensions.

Key ideas

Force and potential energy are linked. If you know one, you can find the other by integrating or differentiating, and the minus sign is part of that link. 10:27
A conservation law lets you skip the step-by-step force calculation. For a spring, you can balance kinetic and potential energy to find the speed at another point. 10:27
Friction cannot be given a single potential energy because its direction depends on how you are moving, not only on where you are. It always opposes motion. 11:33
The same energy idea works for a roller coaster, and in two dimensions the work done depends on the path taken, so the result is harder to use. 14:00
The work-energy idea still holds in two dimensions, but the work depends on the path, so it cannot be written as a simple function of the start and end points. 19:39

Words to know

potential energy: Stored energy that depends on where something is, like a spring that is stretched or an object raised high.
kinetic energy: The energy an object has because it is moving. It depends on mass and speed.
conservative: A force is conservative when it depends only on position, so you can define a potential energy for it and the energy stays balanced.
dot product: A way to multiply two vectors (things with both size and direction) that gives a single number, used here to find work in two dimensions.

Example

Think of a ball rolling over a hill. If you know how high the ball starts and how high it ends, you can work out its speed at the end without tracking every bump in the way. That is the idea of a conservation law: you only need the two endpoints.

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Part 3 · 20:00–30:00

This part explains what makes a force "conservative" (its work depends only on start and end points, not the route), shows a force that fails this test, and gives a recipe for building forces that pass it.

Key ideas

Energy conservation only works if the work done by a force depends only on where you start and where you end. Otherwise you cannot write the energy as a simple difference between two points. 20:11
The force F = i·y is a clear counterexample. Going up first then right gives work 1, but going right first then up gives work 0, so the path matters. 24:12
An equivalent test is that the work around any closed loop is zero. A conservative force does no net work on a trip that returns to where it started. 26:39
You can build conservative forces by picking any function U and setting the force equal to the negative of its derivatives. This is written as F = -grad U. 28:12
The gradient is a machine that takes a function of x and y and returns a vector made from its two derivatives. 29:51

Words to know

conservative force: A force whose work depends only on the starting and ending points, not on the path taken between them.
line integral: A way of adding up small pieces of work along a path. You take the force dotted with each tiny step and sum them all up.
closed path: A path that starts and ends at the same point, like walking a loop and coming home.
gradient: A tool that takes a function and gives back a vector made of its slopes in each direction, such as the x and y directions.

Example

Think of walking up and down a hill and coming back to where you started. For a conservative force, the total work over the full loop is zero, just as your height change is zero when you return to the same spot. This is only an analogy to help the idea click; the lesson does not discuss hills.

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Part 4 · 30:00–40:00

This part explains why some forces always give the same energy result no matter which path you take, and how those forces come from a potential function U.

Key ideas

A conservative force gives the same total result whichever path you take between two points. The result depends only on the start and end points. 30:26
The lecture says every conservative force can be written as the negative gradient of some function U. The lecturer does not prove this second claim. 30:56
When you add up the small pieces of force times distance along a path, you get the change in U between the two points. So the total depends only on the endpoints. 35:31
Because the work equals the change in kinetic energy, kinetic energy plus U stays the same. This is conservation of energy, with U as the potential energy. 37:29
A key question is how to tell if a given force is conservative. One way is to see whether it is the gradient of some function U. 39:32

Words to know

gradient: A set of rates of change, one for each direction (x and y here), packed into one arrow. It tells you how fast a function rises in each direction.
conservative force: A force whose total effect between two points depends only on where you start and where you finish, not on the route.
potential energy: Stored energy that depends on where something is. The lecture calls U the potential energy for these forces.
derivative: A measure of how fast a quantity changes when you change one thing a tiny bit.

Example

Think of U as the height of a mountain. Your change in elevation from the bottom to the top is the same whether you take a steep short trail or a long winding one. Only the starting and ending heights matter.

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

Part 5 · 40:00–50:00

This part explains how to tell whether a force is conservative, and why checking electric forces can be simplified by superposition.

Key ideas

A force is conservative if the energy it stores depends only on the start and end points, not the path. A simple test in two dimensions is to check whether the cross derivatives of the force are equal. 41:55
In general, the work a force does depends on the path taken, so the work-energy theorem alone does not give a law of conservation of energy. Forces that depend only on the end points are the special ones we need. 43:29
Conservative forces are exactly the forces that come from taking the gradient of some potential function U. You can build such forces by starting with any function U. 44:03
Gravity near the Earth is a simple example: its potential is mgz, and its force is -mg pointing down. 45:07
Checking every possible arrangement of electric charges would be impossible. But superposition means that if the test works for one charge, it works for any arrangement of charges. 49:30

Words to know

conservative: A force is conservative when the energy change it causes depends only on where you start and where you finish, not on the route you took.
gradient: A way of collecting all the slopes of a function in each direction into one arrow that points toward the fastest increase of the function.
line integral: A way of adding up a force along a path, counting only the part of the force that points along the path, to get the total work done.
superposition: The idea that the total effect of several charges is simply the sum of the effects of each charge on its own.

Example

Think of carrying a bag up a hill. The energy you store depends only on how high you climb, not on whether you took a steep path or a winding one. That is how gravity works in this lesson: the potential is mgz, so only the height z matters.

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

Part 6 · 50:00–1:00:00

This part shows why the work done by an electric field does not depend on the path taken, and how that leads to electric potential and potential energy.

Key ideas

The field from many charges is just the sum of the fields from each charge. So if you can show the path doesn't matter for one charge, you can show it for all of them. 50:00
Only movement away from the charge, along the field direction, does work. Moving around a circle is at right angles to the force, so it does no work. 52:57
A test charge feels a force equal to its charge times the field. If the field is conservative (path doesn't matter), the force is too, because multiplying by a number does not change that. 53:19
The potential difference between two points is found by taking the minus of the line integral of the field along any path you like. The professor uses a path that goes out radially and then along an arc. 55:38
Potential is the work per unit charge, and potential energy of a charge q is q times the potential. The two words mean different things, and the professor stresses keeping them apart. 59:29

Words to know

conservative: A field is conservative when the work it does going from one point to another is the same no matter which path you take.
superposition: Adding up the effects of several sources, such as adding the fields from each charge to get the total field.
line integral: A way of adding up tiny pieces of a field along a path, counting only the part of the field that points along the path.
potential: The energy per unit charge at a point, so how much work it takes to move a single unit of charge there.

Example

Think of walking up a hill. Your height change depends only on where you start and where you end, not on the winding path you take, as long as you don't go back down. The electric field works in a similar way when the field is conservative.

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

Part 7 · 1:00:00–1:12:10

This part shows how the electric potential, a single number at each point, makes it easier to find electric fields and to track energy, using a single charge and then a dipole as examples.

Key ideas

The potential V at a point can be given any constant offset. The usual choice is to set V to zero when you are infinitely far away, which makes the numbers simple. 1:00:17
Taking the derivative of the potential gives the electric field. The force from a single charge, and from many charges added together, is conservative, which means it follows a conservation of energy rule. 1:01:27
Energy is conserved when you add the kinetic energy (half mass times speed squared) to the charge times the potential at each point. The total stays the same from one point to another. 1:03:47
The potential is easier to work with than the field because it is just a number at each point. You add the numbers from each charge, with no arrows to combine, and then take derivatives to find the field. 1:05:41
For a dipole, two equal and opposite charges close together, the potential far away simplifies to a formula using the dipole moment. This only holds when the distance you are looking from is much bigger than the separation between the charges. 1:10:48

Words to know

potential: A number at each point in space that tells you how much energy a charge would have if it were placed there.
conservative: A force where the energy stays the same in total. The work done only depends on where you start and end, not the path you take.
dipole moment: A measure of a pair of equal and opposite charges, found by multiplying the charge by the distance between them.
gradient: A way of taking derivatives in all directions at once, which turns a potential into the electric field.

Example

Think of a map where every spot has a height number, like a hill map. You can read the heights by adding up numbers, with no arrows needed. The steepness and direction of the slope at any spot then tells you which way things roll, just as the derivative of the potential gives the electric field.

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

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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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