Electric Fields, Charge Distributions, and the GRE

Reviewing for the GRE physics subject test, I've frequently come across questions like:

Given a distribution of discrete charges, (as shown in the diagram), determine the electric field at a given point r, (also shown in the diagram).



Formula for the Electric Field from a Distribution of Charges
The distribution of charges I've seen in sample questions have, so far, involved only finite numbers of point charges, so the electric field equation can be written as a sum:

where

is the distance from a charge contributing to the sum to the location where the electric field is being measured, q is the value of the particular charge, and u is the unit vector pointing from the contributing charge to the field measurement point.

This can be separated into x and y components:

where theta is the angle between the particular charge and the point where the field is being measured with respect to the x axis. Using these formulas, it's just a matter of summing up the contribution of each point charge to the electric field at the location defined by the problem.

So, the standard method is to determine the x and y contribution from each charge at the location specified by the problem, and then simply sum up all the contributions. Let's try one out:

Example

First, we'll try the example from the diagram above:
The contributions from each charge are shown in the table below:



Here's one more example with three equal charges space equally around the unit circle:

The angles used to calculate the x and y contributions of each field are also shown.

The contributions from each charge are shown in the table below:



The Trick
Both of these examples also point out a type of 'trick' question that appears in the samples. When a set of equal charges is evenly spaced around a circle surrounding the field measurement point, the total field is 0. Watch for this pattern.

Handy Stuff
GRE Physics Sample Test [pdf]
Other links to GRE sample questions