Problem 36.

A cube of edge a carries a point charge q at each corner. Find the resultant electric force on any one of the charges.

Solution

We labels four identical charges as: Charge 1, Charge 2, Charge 3, and Charge 4 (as shown in the figure). We calculate the net electric force on charge 1.

There are three electric forces acting on charge 1:

1. due to charge 2. The magnitude of the force is

The distance between the charges 1 and 2 is . Then the force is

2. due to charge3. The magnitude of the force is

The distance between the charges 1 and 3 is . Then the force is

3. due to charge4. The magnitude of the force is

The distance between the charges 1 and 4 is . Then the force is

All these forces are vectors. The directions of the forces are shown in the figure. To find the net force – the vector sum of all three forces – we need to introduce the coordinate system: x and y.

Then the x and y components of the net force are

From the known magnitudes and the directions of the forces we can find the corresponding components:

Then the magnitude of the net force is