Electric fields

Electrostatic repulsion of a person's hairs. Biswarup Ganguly, CC BY 3.0 <https://creativecommons.org/licenses/by/3.0>, via Wikimedia Commons

Electrostatic attraction of styrofoam peanuts to a cat. Sean McGrath, CC BY 2.0 <https://creativecommons.org/licenses/by/2.0>, via Wikimedia Commons

Forces of attraction and repulsion

Pictured above are two examples of forces created by electric fields: electrostatic repulsion among the hairs of a person, and electrostatic attraction between styrofoam peanuts and a cat. These kinds of forces have been known since antiquity. Ben Franklin (yes, the Declaration of Independence signer) did ingenious experiments to demonstrate the existence of electrical charge as the source of these forces. Here is a picture of his electrostatic machine. By turning a crank, he could spin a glass globe and cause it to rub against a leather pad underneath it. The glass would gain a positive charge and the pad a negative charge. Franklin proposed that this was because a positively charged fluid was being transferred from the pad to the glass. This would leave a deficit of the fluid in the pad, creating an equal amount of negative charge. This was a breakthrough because he realized that charge is only transferred, not created or destroyed. Only one type of charge needs to flow from one body to another to have them end up having opposite charges.

Franklin's electric charge generator. Flickr: Adam Cooperstein, CC BY 2.0 <https://creativecommons.org/licenses/by/2.0>, via Wikimedia Commons

Franklin found that either type of charge could be transferred to other objects, causing sparks and causing shocks when those other objects are human bodies.

He also observed forces of attraction and repulsion. Like charges repel each other and opposite charges attract. Newton's Third Law tells us that in either case, the forces between a pair of charges are equally strong and in opposite directions. Charles Coulomb later measured these forces and found that the attraction or repulsion is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

An object with a large positive charge and one with a small positive charge exert equal and opposite forces on each other

The idea of a field

It's interesting that charges can attract or repel each other from a distance without even touching. This inspired the idea of an electric field, which is an influence that a charge creates in the space around it, visualized as a set of lines emanating from it. The larger the amount of charge, the more lines it produces.

Electric field lines emanating from a positive charge

These lines show the direction of the force that the charge would exert on a second positive charge. The field created by a negative charge would look the same, except that we'd draw the arrowheads pointing inward, because that second charge, if it were positive, would be attracted rather than repelled.

This representation of the electric field has another nice feature. The strength of the field is represented by how closely spaced the lines are. As one goes farther away from a charge its field lines get farther apart, which means that the field is getting weaker. This works not only as a qualitative description, but also works quantitatively. A given charge produces a fixed number of lines. Mathematically, this causes the density of the lines to decrease as the square of the distance, in agreement with Coulomb's measurements of the forces.

The angry porcupine theorem

How do we draw the field created by multiple charges in different locations? In 1865, James Clerk Maxwell published his now famous equations of electromagnetism. One of these equations, known as Gauss's Law, relates the electric field to the charges producing it. I have a humorous way of introducing it to students as the "angry porcupine theorem."

The angry porcupine theorem

Imagine an unknown number of angry porcupines hidden in a paper bag. They are so angry that they have thrown all their quills through the bag, even at the risk of impaling each other. We want to know how many porcupines are in the bag without taking the risk of looking inside. If we know how many quills a porcupine has, then we just count the number of quills we see and then divide by that number. For example, if a porcupine has 10,000 quills, and there are 30,000 quills protruding from the bag, we know that there are three porcupines inside.

Think of a unit of positive charge as being like a porcupine and the electric field lines being like quills coming out of it But now we also need to include negative charges, whose electric field lines come into it. We can create a mathematical paper bag, and relate the net number of electric field lines coming out of the bag to the net amount of positive charge inside it. I say "net" because when we use Gauss's Law, we take the number of lines coming out minus the number going in, and we relate that to the net charge inside, positive minus negative.

In the example pictured below, the net charge inside the gray region is positive because there is a large positive charge incompletely cancelled by a small negative charge. Accordingly we find that the net number of lines coming out of the gray region is positive; there are more lines coming out than going in.

In this example, the number of electric field lines coming out of the gray region is greater than the number going in. The net charge inside must be positive. MikeRun, CC BY-SA 4.0 <https://creativecommons.org/licenses/by-sa/4.0>, via Wikimedia Commons

Summing it up

Electric field lines start on positive charges and end on negative charges. They tell us at each point the direction of the force that would be experienced by an added positive charge. The more closely spaced the field lines, the stronger the force. Gauss's Law expresses these relationships and enables us to calculate the electric field pattern produced by a collection of charges.