Lecture notes · Physics: Electricity & Magnetism · Lecture 2

Electric Fields

Yale PHYS 201 Fundamentals of Physics II, lecture 2, explained in plain English

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Part 1 · 0:00–10:00

This part explains how electric charges push and pull on each other, how forces from many charges add up, and why protons stay together in a nucleus even though they repel each other.

Key ideas

Electric charge is a property some particles have, measured in coulombs. Electrons have a negative charge, protons have a positive charge, and neutrons have no charge. 1:07
Two charges pull or push on each other with a force that gets weaker as the distance grows, following a 1 over r squared rule. Charges with the same sign repel, and charges with opposite signs attract. 1:51
When several charges are present, you find the force from each one separately and add those forces as vectors (arrows that have both size and direction). Each pair acts as if the others were not there. 3:58
Protons in a nucleus repel each other, so something stronger must hold them together. That is the strong nuclear force, which is stronger than the electric force. 7:23
The strong nuclear force has a short range. Its strength drops off very quickly with distance, so it only wins over electric repulsion when particles are very close together, inside the nucleus. 8:35

Words to know

electric charge: A basic property of some particles that makes them push or pull on other charged particles. It is measured in coulombs.
coulomb: The unit used to measure electric charge, like how meters measure length.
superposition principle: The rule that the total force on a charge is found by adding up the separate forces from each other charge, as arrows.
range of the force: How far a force reaches. A force with a short range gets very weak quickly as you move away from its source.

Example

Think of several people pushing on the same box from different sides. You can work out where the box goes by adding up each person's push as an arrow, and each person's push is the same whether or not the others are pushing. This is like how superposition lets you add up the electric forces from many charges.

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Part 2 · 10:00–20:00

This part explains why nuclei have a size limit, how the Coulomb force is written as a vector, and why physicists use the idea of an electric field.

Key ideas

Neutrons add attraction between protons through the nuclear force without adding to the electric push between protons. That is why they act like glue inside a nucleus. 10:16
The electric repulsion between protons falls off slowly over distance, so in a very large nucleus it wins over the short-range nuclear attraction. This is why nuclei cannot grow beyond a certain size. 11:14
The Coulomb force is a vector, so it has a size and a direction. You write it as a strength times a unit vector that points from one charge to the other. 14:22
An electric field is a value that exists at every point in space, set up by a charge even when no other charge is there. A charge placed in that field feels a force equal to the field times the charge. 17:36
The electric field at a point can be thought of as the force a unit charge would feel if placed there. You can measure it by putting a small test charge at that spot and seeing what force it feels. 19:07

Words to know

Coulomb's Law: A rule that says how strongly two electric charges push or pull on each other. The force depends on the size of the charges and how far apart they are.
Nuclear force: A very strong pull that holds protons and neutrons together inside a nucleus. It only works over very short distances.
Unit vector: An arrow with a length of exactly one. It only shows direction, and multiplying it by a number gives an arrow of that length pointing the same way.
Electric field: An invisible influence that a charge spreads through the space around it. Any charge placed in that space feels a push or pull from it.

Example

Think of a fan blowing air through a room. The air is moving at every spot in the room, whether or not anything is there to be pushed. An electric field works the same way: it is present at every point, and a charge that enters that point feels its push.

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Part 3 · 20:00–30:00

This part explains how to find the electric field at a point, how a charge feels that field, and why the simple Coulomb's Law formula breaks down when charges move.

Key ideas

To measure the field at a point, place a small test charge there and see how much it accelerates. Multiply that acceleration by the mass to get the force, then divide the force by the charge to get the field. 21:04
One charge is enough to create a field. Its presence changes the space around it, even if no second charge is there. 21:24
The total field at a point is the sum of the fields from all the charges, added as vectors (quantities with both size and direction). 22:41
Coulomb's Law is only an approximation when charges move. Effects travel at the speed of light, so the field at a point depends on what charges did in the past. 24:49
Modern physics splits the force into two steps. Charges produce a field, and other charges respond to the field at their own location, not to the charge itself. 27:27

Words to know

test charge: A small charge you place at a spot to find out what forces are there. It is used only to measure, not to create the field you are checking.
electric field: The force that a charge of 1 unit would feel at a point. Every charge creates one, and it is present even when no other charge is there to feel it.
Coulomb's Law: The formula that gives the force between two charges that are not moving. The lesson says it is good for still charges but only an approximation for moving ones.
special theory of relativity: A theory that says no signal, and no influence, can travel faster than light. The lesson uses it to show why instant communication between charges is not allowed.

Example

Think of a magnet's pull on a paper clip. You can measure the pull by holding a paper clip at a spot and seeing how hard it is yanked, even though the magnet is what creates the pull. The paper clip is like the test charge: it tells you about the field without being the cause of it.

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

Part 4 · 30:00–40:00

This part explains how to draw electric field lines so a picture shows both which way the field points and how strong it is.

Key ideas

An electric field arrow is drawn at one point, and it describes the field only at that point, not along the arrow's whole length. The field is like a contour map: it shows what is happening at each spot. 30:20
If you join many arrows into continuous lines, you can see the field's direction everywhere, but you lose the information about how strong the field is. 32:40
The strength of the field is shown by how crowded the lines are. Where lines are close together the field is strong, and where they are spread apart the field is weak. 34:52
The lines spread out as you move away from a charge, so their density falls off with distance in the same way the force does. This works because we live in three dimensions. 36:06
By agreement, one coulomb gets 1 over epsilon-zero lines. With this choice, the number of lines crossing a unit area equals the field strength, so you can measure the field by counting lines through a piece of wood one meter square. 37:59

Words to know

Electric field: A property at each point in space that tells you what force a charge placed there would feel, including its direction and strength.
Test charge: A small charge you place in a field to find out what force the field puts on it.
Coulomb: The unit used to measure electric charge.
Density (of lines): How many lines pass through a surface, divided by the size of that surface. More lines in a small area means higher density.

Example

Think of a hiking contour map. Each line marks a height, and where the lines are packed tightly the ground is steep. Field lines work the same way: where they are packed tightly, the electric field is strong.

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

Part 5 · 40:00–50:00

This part explains how to picture electric fields around pairs of charges, and it starts a calculation of the field of a dipole, which is two opposite charges close together.

Key ideas

Close to a charge, that charge's field dominates, because the field gets very strong as you get very near it. Far-off charges matter much less at that spot. 40:19
Field lines point away from a positive charge and into a negative charge. A test charge is a small positive charge used to check the field, and it would be pushed or pulled along those lines. 40:59
Two opposite charges close together make a dipole. Its field lines start at the positive charge and end at the negative one, and they are hard to draw exactly, so a good sketch is often enough. 41:55
From very far away, a group of charges looks like one single point charge with the total (net) charge, and the lines spread out evenly from it. 43:05
To find the dipole's field on the line through both charges, you add the pull of each charge. Combining the two terms gives a result of the form q over 4 pi epsilon-zero, times 4xa over (x squared minus a squared) squared. The field is zero when the two charges sit on top of each other (a = 0). 48:15

Words to know

field: A way of describing the push or pull that a charge would feel at each point in space.
dipole: Two charges of equal size and opposite sign that sit close together.
test charge: A small positive charge imagined at a spot, used to find out which way the field pushes.
field lines: Drawn arrows that show the direction a test charge would move in the field.

Example

Think of a tug-of-war where two people pull a rope in opposite directions. Up close, you can see each person pulling separately, but from far away the rope just seems to be pulled one way. A dipole behaves in a similar way: close up you see two separate charges, and from far away you see one combined effect.

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

Part 6 · 50:00–1:00:00

This part shows how to find the electric field of a dipole, a pair of equal and opposite charges, and how a charged particle moves in a constant electric field.

Key ideas

Two equal and opposite charges placed close together make a dipole. Far from it, the field is much weaker than near it, and it falls off as 1 over x cubed. 50:10
The dipole moment p is the charge times the small vector between the two charges. It points in the same direction as the field far along the axis. 52:09
A single charge's field falls as 1 over x squared. A dipole has zero net charge, so its field falls faster, because the two charges nearly cancel each other out. 52:51
On the line that cuts the dipole in half, the sideways parts of the two fields cancel, leaving a field of minus p divided by 4 pi epsilon zero times r cubed. 55:39
A particle moving between two charged plates with a constant field gets a constant sideways push. Its path is a curve: it moves steadily forward and falls more and more. 58:53

Words to know

dipole: A pair of charges that are equal in size and opposite in sign, placed close together.
dipole moment: A number and direction that describe a dipole. It is the charge times the distance between the two charges.
vector: A quantity that has both a size and a direction, such as a push or a velocity.
acceleration: How quickly an object's speed or direction changes over time.

Example

Think of a magnet with a north end and a south end. Stand far away and it looks weak and simple, but the pull drops off faster than it would for a single charge. The two ends mostly cancel each other out when you are far away.

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

Part 7 · 1:00:00–1:12:25

This part explains what happens to an electric dipole, a pair of equal and opposite charges, when it sits in a uniform electric field: it feels no overall push, but it gets twisted until it lines up with the field.

Key ideas

In a uniform electric field, the two charges of a dipole are pulled in opposite directions with equal strength. So the dipole gets no overall push, even though each charge is pulled. 1:02:15
Although there is no net push, there is a torque, which is a twisting force. The torque is largest when the dipole is at a right angle to the field, and it is zero when the dipole points along the field. 1:03:23
A dipole in a field acts like a spring that wants to return to its resting position. The energy linked to its angle is written as minus p dot E, where p is the dipole moment and E is the field. 1:05:12
Lining up with the field is a stable resting spot, like a marble at the bottom of a bowl. Pointing against the field is unstable, like a marble balanced on top of a hill. 1:08:31
The big picture is that charges create electric fields, and fields push on charges. Far from two equal and opposite charges, the field gets weaker very fast, falling off as 1 over r cubed. 1:10:27

Words to know

dipole: A pair of charges that are equal in size and opposite in sign, placed a small distance apart.
torque: A twisting force that tries to make something rotate, like turning a screwdriver or a wrench.
cross product: A way of multiplying two arrows (vectors) to get a new arrow. Its size depends on the angle between them, and its direction follows the screwdriver rule.
potential energy: Stored energy that depends on where something is. A ball held high has potential energy that can turn into motion when it falls.

Example

Think of a marble resting in the bottom of a bowl. If you nudge it, it rolls back and forth but comes to rest in the middle, which is a stable spot. A marble balanced on top of a hill is in an unstable spot, because the slightest push makes it roll away. A dipole in an electric field behaves in a similar way: lined up with the field it settles down, but pointed against the field it can flip over.

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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