Recall that work done on an object is the negative of the change in potential energy of the object. Work is the energy that is tranferred when an object is displaced by a force.

For a constant force, work is defined as the dot product between the force on the object and the displacement of the object. If the force is not constant, we need to integrate over the displacement to find the work done.

In general, we take the integral of the force over the displacement to find the total work done. We can use this formulation to calculate the work done on a charged particle by another charged particle over a displacement. We use the relationship between work and potential energy to find the change in potential energy of the charged particle.

For the potential energy of two charged particles, we cah consider the initial distance between the particles to be infinite. In this limit, the initial potential energy of the system equals zero.

For a system of point charges, we add the potential energy from each pair of charges to get the potential energy of the system. For example, a system of three point charges will have contributions from each pair, giving three terms to the sum. This is a scalar equation, so the terms simply add.

Example problem

Find the potential energy of this system of point charges.

Use the definition of the total potential energy.

The idea here is that it takes work to bring the charges together. The first particle comes in from infinity for free. Since there are no other charges present yet, there is no force. The first particle exerts either an attractive or repulsive force on the second particle as it is displaced. The first and second particle exert forces on the third particle as it is displaced. The total potential energy of the system is the negative of the total work done.

Sample problems

1. Find the potential energy of this system of point charges.

2. A line of constant continuous charge, with total charge Q, extends from -L/2 to L/2 along the x-axis as shown. Find the potential from the line of charge at a point P a distance d on the x-axis.

3. A wire of constant continuous charge, with total charge Q, is shaped in a semicircle of radius R, centered on the origin as shown. Find the potential from the wire at the origin.