# Differentiation

5.21: Electric field of a charge distribution:
Suppose we have a distribution of charges and we want to calculate the resulting electric field. One way to do this is to first calculate the electric potential φ and then take its gradient. For a point charge q at the origin, the electric potential at a distance r from the origin is

$\Phi = {q\over{4\pi\epsilon_0 r}}$

and the electric field is E = −∇φ.

1. You have two charges, of ±1 C, 10 cm apart. Calculate the resulting electric potential on a 1 m × 1 m square plane surrounding the charges and passing through them. Calculate the potential at 1 cm spaced points in a grid and make a visualization on the screen of the potential using a density plot.
2. Now calculate the partial derivatives of the potential with respect to x and y and hence find the electric field in the xy plane. Make a visualization of the field also. This is a little trickier than visualizing the potential, because the electric field has both magnitude and direction. A visualization might use the arrow object from the visual package, drawing a grid of arrows with direction and length chosen to represent the field.