For the next two weeks we are going to investigate the electric potential. This week we are going to study equipotential lines, gradients, and the relation to the electric field. Next week we are going to explore the dynamics of charge motion in an electric potential. The electric potential is easier to use than the electric field because the potential is a scalar field. By Coulomb's convention for a point charge, V = kq/r, positive charge creates hills, and negative charge creates valley's.
The electric field can be found from the potential by taking the negative gradient of the potential. The gradient is the generalization of the slope of a graph to 3-D. If we think of the potential as a hill, the gradient always points uphill in the direction of positive slope.
The electric field is opposite to the gradient, E = - grad V, and can be thought of as flowing down the potential hill.
Next week we will see that if a positive charge is placed in this potential it experiences a force, F = qE , or in terms of the potential, F = - q grad V. Positive charge will flow downhill. If we place a positive test charge near another positive charge Q, then Q will create a potential hill with Q at its center. The test charge will move downhill, away from Q, and hence, the force on the test charge will be repulsive. On the other hand negative charge creates a hole or valley with -Q at its center and into which the test charge will tend to fall. Thus, the test charge will be attracted by negative charge.
Next review superposition for waves.