Basics: The electric field

Suppose I take a 1 kg ball and hold it near the surface of the Earth. What would be the gravitational force the Earth exerts on this ball?

And I could say "g" is:

The magnitude of this force would then be 9.8 Newtons. And, if I replaced the ball with a 10 kg ball, the force would be 98 Newtons. What does this have to do with the electric field? Well, you are probably already familiar with this idea of the gravitational force. Guess what? "g" is the gravitational field. Basically, it is the force per unit mass due to the Earth. This is only approximately constant. If I get very far from the surface of the Earth, it might be better to write the gravitational field as:

This is the universal gravity force divided by the mass of the object at the location of interest. So, you could say take a small "test mass" and determine the force on it. If you divide the force on the test mass by the value of the test mass, you get the gravitational field. (note - r-hat points away from the surface of the Earth and G is the gravitational constant.) In general, the gravitational field is:

Maybe you see where this is going or maybe you are thinking "hey, this is supposed to be about the electric field". Yes, it is supposed to be about the electric field. Suppose I put some electric charge somewhere and measure the electric force on it. In that case, I can say the electric field is:

What about a point charge? If I take some electric charge - q₁ and I want to find the electric field a distance r away. I will put a small "test charge" there with charge q_t. The force on that charge is:

If I divide this force by the value of the test charge, I get the electric field due to a point charge.

Superposition

The electric field due to a point charge is useful because stuff is made up of things that look like point charges - (electron, proton). But what about other things? What about the electric field due to two point charges? The cool thing about the electric field is that it obeys the idea of superposition. This means that the electric field at any point due to two point charges is the vector some of the electric fields due to each individual charge. Here is a diagram.

It doesn't matter how many charges you have. You can keep adding up these electric fields due to the individual point charges. This is quite useful - as I will hopefully show later.