Lecture notes · Physics: Electricity & Magnetism · Lecture 3

Gauss's Law I

Yale PHYS 201 Fundamentals of Physics II, lecture 3, explained in plain English

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Part 1 · 0:00–10:00

This part explains the electric field, how field lines show it, how its strength drops with distance, and what happens to a dipole placed in a steady field.

Key ideas

The electric field at a point is the force a 1 coulomb test charge would feel there, found by adding up the pushes and pulls from all the charges. Even with nothing at the point, the field is real because it acts on anything placed there. 0:23
If you know the electric field at a point, any charge q placed there feels a force equal to q times E. The field tells you the force for any amount of charge. 1:29
Field lines show the direction of the force at each point. The line density, meaning how many lines pass through a square meter held perpendicular to them, shows how strong the field is. 2:13
The field from one point charge falls off as 1/r squared, and the field from a dipole falls off faster, as 1/r cubed. Distance makes the field weaker quickly. 6:18
A dipole placed in a uniform field feels no net force, because the push on each end cancels out. It can still react in other ways. 9:42

Words to know

electric field: An invisible influence at every point in space. If you put a charge there, the field pushes or pulls on it with a force you can calculate.
coulomb: The unit used to measure electric charge. The lesson imagines putting 1 coulomb at a point to measure the field there.
dipole: Two equal and opposite charges, one positive and one negative, placed near each other. Think of a tiny pair of charged ends.
area density: How much of something passes through a given area. For field lines, it means counting how many lines cross a 1 square meter frame held across them.

Example

Think of a weather map where every spot has an arrow showing which way the wind blows. Closely packed arrows mean strong wind, and spread-out arrows mean weak wind. Field lines work the same way for electric fields.

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

Part 2 · 10:00–20:00

This part explains how a force can turn a dipole (a pair of opposite charges) and how a force links to potential energy, then starts a problem about the electric field around an infinite line of charge.

Key ideas

A dipole in an electric field feels a torque, which is a twisting push that tries to line it up with the field. The torque comes from the force times the distance and the sine of the angle between them. 10:00
Only the part of the force that is perpendicular to the axis can rotate the dipole. The part along its length just tries to stretch it, so it does not twist it. 10:29
If a force is not friction, you can connect it to a potential energy. The force is found from the energy, and the energy is found by adding up the force over a distance. 11:39
A potential energy is only defined up to a constant. Adding the same number to every value does not change any difference in energy, so we pick the constant that keeps the formula simple. 12:53
For a line of charge that goes on forever, symmetry tells you the field must point straight out from the line. It cannot lean left or right, because the wire looks the same in both directions. 16:28

Words to know

torque: A twisting push that makes something turn around a point. It depends on how hard you push and how far from the turning point you push.
dipole: A pair of equal and opposite charges that sit close together. Think of a tiny magnet with two ends.
potential energy: Stored energy that depends on where something is. A stretched spring holds potential energy, and that energy depends on how far it has stretched.
dot product: A way to multiply two arrows (vectors) that gives a single number. The lesson uses it to write the potential energy of a dipole as p times E times the cosine of the angle.

Example

Think of a door. Pushing near the hinges does little to turn it, but pushing straight across the door, far from the hinges, turns it easily. Only the part of your push that is at a right angle to the door does the turning, just as only the perpendicular part of a force turns the dipole.

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

Part 3 · 20:00–30:00

This part works out the electric field from a very long charged wire, then starts on the field from a flat sheet of charge.

Key ideas

Only the part of each tiny piece's field that points straight out from the line adds up, so the professor takes the cosine of the angle to get that part. 20:00
The function is even, so the positive side and the negative side of the wire give equal results. You can calculate one half and double it. 20:59
Before doing the math, you can use dimension analysis, which checks units, to predict the answer's shape: it must be a number divided by the distance squared. 23:28
A trig substitution, x = a tan θ, turns the hard integral into one you can solve, and the result comes out to a simple number. 26:07
The field from an infinite wire weakens with distance as 1 over distance, not 1 over distance squared. For the sheet, the field does not depend on how far you stand in front of it. 27:51

Words to know

even function: A function that gives the same result when you flip the sign of its input, like x becoming minus x.
integrand: The expression inside an integral sign that you are adding up over a range.
dimension analysis: Checking the units on both sides of an equation to see what shape the answer must have.
charge density: How much electric charge is packed into each unit of space, such as coulombs per square meter for a sheet.

Example

Think of a very long line of street lamps. One lamp on its own looks dim from far away, but a line that goes on forever adds up to light that fades more slowly than a single lamp would. The total depends on the whole line, not just one lamp.

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

Part 4 · 30:00–40:00

This part shows how to find the electric field of a flat, endless sheet of charge by using symmetry and then adding up thin rings of charge.

Key ideas

An infinite sheet looks the same if you slide it sideways or turn it, so the field must also look the same after those moves. That means the field points straight out from the plane, not at a tilt. 30:14
The field is the same strength at every point the same distance above the plane. You do not need symmetry to get the answer, but it tells you what to expect, so you can spot mistakes. 31:49
To do the math, the speaker splits the sheet into thin rings. For each small piece of a ring, sideways parts cancel out with matching pieces on the other side, so only the straight-out part is left. 34:15
Every piece of a ring is the same distance from the point, so you just add up the ring's area. A thin ring's area is 2πr dr, and you add all the rings by integrating over r from 0 to infinity. 35:51
The integral is easier with the substitution w = r squared. The r on top lets you swap the whole top part for dw, which turns the problem into a simple power integral. 39:07

Words to know

symmetry: a property where something looks the same after you move or turn it. Symmetry arguments use this to predict how a field must behave, without doing much math.
annulus: a flat ring shape, like a thin washer, with a inner edge and an outer edge. Here it has radius r and a very small width dr.
integrate: to add up a huge number of tiny pieces to get one total. Here the tiny pieces are thin rings, and you add them all from the center out to infinity.
substitution: replacing a messy part of a math problem with a new letter, so the problem becomes easier to solve. Here w = r squared is the swap used.

Example

Think of a huge flat sheet of frosting spread evenly over a table that goes on forever. If you slide or spin the sheet, nothing about it seems different, so the push it gives you at any spot above it must point straight out of the sheet, with the same strength everywhere at the same height.

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

Part 5 · 40:00–50:00

This part introduces the infinite charged plane, explains why a charged sphere is hard to solve directly, and teaches how a flat area can be written as a vector.

Key ideas

For an infinite flat sheet of charge, the electric field stays the same no matter how far you move away from it. The result is sigma over 2 epsilon-zero. 40:21
Each piece of the sheet gets weaker with distance, yet the total field does not fall off. You only learn this for certain by doing the full calculation. 41:52
Finding the field of a solid charged ball by adding up every tiny piece is very hard, because each piece sits at a different distance and angle. The lesson says Gauss's law is a powerful trick for this kind of problem. 44:20
A small flat area can be treated as a vector. Its size is the area, and its direction is perpendicular to the surface, with the sign chosen by the right hand rule. 46:02
Flow rate is the volume of fluid that passes a cross section each second. For a rectangular tube, the cross section area is h times w. 49:51

Words to know

Normal: A line that sticks straight out from a surface at a right angle, like a pole standing straight up from a table.
Charge density (rho): How much electric charge is packed into each cubic meter of a space. It is written with the Greek letter rho.
Cross product: A way to multiply two vectors in three dimensions that gives a new vector pointing straight out from the flat space they make.
Flow rate: How much stuff passes a certain point each second, such as cubic meters of fluid per second.

Example

Picture air blowing through a rectangular air duct. Each second, a certain volume of air goes past any cross section of the duct, and that amount is the flow rate. The lesson uses this same idea to explain why a flat opening can be given a direction.

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

Part 6 · 50:00–1:00:00

Flow through a surface is found by multiplying velocity by area with a dot product, and the total field lines leaving a charge through any closed surface depend only on the charge inside.

Key ideas

The flow rate of a fluid past a cross section is the area times the speed. A faster fluid or a bigger opening lets more stuff through. 50:20
Area is treated as a vector that points straight out from the surface. Its dot product with velocity counts only the part of the flow that actually goes through the area. 51:07
A tilted area is bigger, but it catches no more fluid, because the cosine of the angle cuts the effect back. An area parallel to the flow catches nothing. 52:31
The electric field tells you how many field lines pass through each small patch. Adding up E dot dA over a sphere around a charge gives q divided by epsilon zero, and the answer does not depend on the sphere's radius. 55:31
Because field lines never end in empty space, the count is the same for any closed surface around the charge, even a strange shaped one. This is the surface integral of E equals q over epsilon zero. 58:31

Words to know

Flux: The amount of something that flows through a surface in a given time, such as fluid or field lines.
Dot product: A way to multiply two vectors that gives one number. It depends on the size of each vector and the cosine of the angle between them.
Surface integral: A way to add up a quantity over a surface by splitting the surface into tiny pieces, taking the dot product of each piece with the field, and summing them all.
Cross section: A flat slice through something, like the opening of a pipe, that the fluid must pass through.

Example

Think of holding a hula hoop in a steady rain. If the hoop faces the falling rain straight on, it catches the most drops. If you tilt it so its face is nearly flat, almost no drops go through it. The same idea holds for field lines through a closed surface.

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

Part 7 · 1:00:00–1:10:21

This part introduces Gauss's law, which links the total electric field leaving a closed surface to the charge inside it, and shows how it gives the electric field around a solid ball of charge.

Key ideas

For a closed surface, the total electric field passing out through it equals the charge inside divided by ε0 (a constant number used in electricity). This is Gauss's law. 1:00:22
If the charge sits outside the surface, the net number of field lines leaving is zero, because lines that go in must also come out. Where the charge sits inside the surface does not matter. 1:01:15
When charge is spread out smoothly rather than in separate pieces, you add up the density at every tiny cube inside the volume to get the total charge enclosed. 1:02:58
For a solid ball of charge, symmetry tells you the field points straight out and has the same size everywhere on a sphere. That lets you pull the field out of the sum and solve for it, giving the same result as a point charge. 1:07:55
Gauss's law only helps you find an answer when there is just one unknown number in the problem, such as the field strength at a given radius. 1:09:50

Words to know

Flux: The total amount of field lines passing through a surface. Think of it as a count of how much flows through.
Closed surface: A surface with no holes or gaps, like the skin of a ball or a potato, that fully surrounds a space.
Charge density: How much electric charge is packed into each small piece of space. A high density means lots of charge in a small place.
Radial: Pointing straight out from the center, like the spokes of a wheel.

Example

Imagine a sealed box with a tap inside that is pouring water into it. Whatever flows out through the box's walls must match what the tap is adding, and a tap outside the box adds nothing to the total leaving it. This is a rough picture of how Gauss's law counts what leaves a closed surface.

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

← Lecture 2: Electric Fields · Lecture 5: The Electric Potential and Conservation of Energy →

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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