Work and potential energy
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 can 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.
The electric potential energy is a property of a system of charges. We can also define an electric potential, which is a field quantity. The electric potential is the sum of the energy per charge for a system of particles, measured at a point in space.
Electric potential of a charged sphere
Outside a uniformly charged sphere, the electric potential is the same as that for a point charge located at the center of the sphere.
Here, we have defined the potential at a distance r from the center of the sphere for a sphere of radius R. It is common to speak of charging a sphere to a potential by placing an amount of charge on it.
We define the potential to be V0 at the surface of the sphere.
We can combine these two equations to find the potential at a distance r from a sphere charged to V0 volts. What about inside the sphere? We know that a conducting sphere has zero electric field everywhere in its interior.
If the electric field is zero inside the conducting sphere, the change in the potential must be constant.
To match the boundary condition at the surface, the electric potential inside a conducting sphere must equal the potential at the surface.
For a non-conducting sphere, the electric field is not zero inside so the electric potential is not constant.
We used Gauss's law to show that the electric sphere is linear in r inside a charged non-conducting sphere with constant charge density.
We can use the expression for the E field to find the electric potential inside a uniformly charged non-conducting sphere.
Performing the integration and applying the boundary conditions gives us an expression for the change in potential.
We can include our expression for the potential at the surface of the sphere.
Simplification gives us a compact expression for the electric potential inside a uniformly charged non-conducting sphere.
Sample problems
1. Find the potential energy of this system of point charges.
2. Find the potential from this system of charges at point P.
3. 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.
4. 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.
5. A nonconducting sphere of uniform charge density charged to V0 is fixed in place. A point charge of mass m and charge q is released from rest at its surface. How fast is the particle moving when it has gone a distance d?
