Very Ancient Electromagnetism Notes

Electromagnetism: electricity and magnetism, two aspects of the same fundamental interaction.

Charge

Charge is a property of things, just like mass is.

There are two types of charge: positive and negative. These are arbitrary labels, and could have been called anything.

Two common "units" of charge:

• The Coulomb () is the SI unit of charge.
• The "elementary charge" () is the charge of a proton or electron. It is approximately .

Electric Force

Electric force is the force between two charged objects. Opposites attract, likes repel.

In 1785, Charles Coulomb discovered that the force between two charges is proportional to the product of the charges and inversely proportional to the square of the distance between them. This is Coulomb's Law, and it is given by:

Where:

• is the force between the charges.
• is Coulomb's constant, approximately .
• and are the charges.
• is the distance between the charges.

Try dragging the points around to see how the force changes (the force is shown by the vectors).

You would notice that the force gets way stronger when the charges are closer together.

Mathematically, this is due to proportionality: .

Materials and Electricity

Electricity is the flow of electric charge. This flow, or "current", is what powers our electronics.

Materials can be classified based on their ability to conduct electricity.

• Conductors: materials that allow electric charge to flow freely. Metals are good conductors.
• Insulators: materials that do not allow electric charge to flow freely. Rubber and plastic are good insulators.

There are also semiconductors and superconductors, but for the purposes of this introduction, we simplify things by ignoring them.

In a conductor, electrons can move freely. In an insulator, they cannot. This is due to the structure of the material. For instance, -block metals have a "sea of electrons" that can move freely, while covalent bonds in insulators hold electrons in place. This is why metals like copper are good conductors, while plastics are good insulators.

However, insulators also have usages in electronics. For instance, putting an insulator in an electric field can cause its nucleus to move, which polarizes the material.

The rightwards arrows represent an electric field. This field acts on the positively charged nucleus by pulling it to the right.

Induction

One way to charge an object without touching it is through induction.

Here we have two rods. The red rod (Rod A) is negatively charged, and the white rod (Rod B) is neutral. This white rod is connected to the ground.

When we bring Rod A closer to Rod B:

1. The negative charges in Rod A repel the negative charges in Rod B, causing them to move to the right.
2. The positive charges in Rod B are attracted to the negative charges in the red rod, causing them to move to the left.
3. The negative charges in Rod B escape to the ground.

This leaves Rod B with a net positive charge, even though we didn't ever touch it.

Conservation of Charge

Like energy and momentum, charge is conserved. Mathematically,

Looks complicated, but these are what the symbols mean:

• is the rate of change of charge.
• is the rate of charge entering a system.
• is the rate of charge leaving a system.

In a system where no charge is entering or leaving, the total charge is therefore constant:

In other words,

This is true even at the quantum level. For instance, when a photon emits an electron, it also emits a positron to conserve charge.

Electric Field

People have asked the question of: "How does charge apply a force over a distance?"

Michael Faraday, a British scientist, came up with the concept of the electric field to answer this question.

The electric field is a vector field that describes the force that a charge would experience at any point in space. This field exists everywhere in space just like the gravitational field.

In the diagram above, the green arrows represent the electric field. The blue arrow represents a test charge. (You need to move the green point a bit for the field to render properly due to a bug.)

A test charge is a charge that is so small that it does not affect the electric field. It is used to measure the electric field at a point.

Technically, test charges don't exist. However, they are a useful concept for understanding the electric field.

Move the blue point around. You would notice that the force on the test charge changes as it moves around.

Mathematically, the electric field at a point is given by:

Where:

• is the electric field.
• is the force on a test charge.
• is the magnitude of the test charge.

hence has units of force per charge, or .

As a vector field, the electric field has both magnitude and direction. If the charge is positive, the field points away from the charge. If the charge is negative, the field points towards the charge.

If you take the magnitude of the electric field, it can be simplified to:

In practice, it is often easier to just get the magnitude and then conceptually find the direction.

Combining Fields

The electric field is a vector field, so it can be added together.

For instance, if we have two charges, the electric field at a point is the sum of the electric fields due to each charge.

In the diagram above, the red and green arrows represent the electric fields due to the red and green charges respectively. The blue arrow represents a test charge.

Now, if we combine these fields into a new field, it looks like this:

The blue arrow represents the electric field at the test charge due to both charges. To visualize this, if you start at each individual point in this grid, here's what it looks like:

This was made by simulating the path of a test charge in the combined field. The path is calculated by taking small steps in the direction of the field at each point. As far as I know, there is no known closed-form solution for this two-charge problem.

Field from Infinite Plate

Imagine a plate that extends infinitely in all directions. This plate has a uniform charge density .

To simplify the problem first, we think of a 1D version. We can find the electric field at a point above the plate. Imagine a test charge that is a distance above the plate. Then, imagine two points on the plate away from the point above the plate. The -component of the field would be , and the -components cancel out. Since cosine is adjacent over hypotenuse, we can use trigonometry to find the field at the point above the plate.

Now we do this in 2D: instead of two points, we imagine a ring of radius and thickness on the plate. The charge can be written as area times charge density, which is:

The force experienced by is:

And the electric field is hence:

Like the 2D version, we take the -component because the horizontal components cancel out.

This is just for a ring. To get the field at the point above the plate, we integrate over all rings.

We can use a substitution to solve this integral.

So the integral becomes:

This is the electric field at a point above an infinite plate. The interesting thing is that this doesn't depend on at all. Hence, this field is uniform - it is the same at all points above the plate.