We will start with a sphere of
radius a that already carries charge q. We want to determine the
work it will take to move an additional small amount of charge dq from
infinity to the surface of the sphere.
We've figured out how much work
would be done moving dq to the surface of the sphere. Some work
had already been done to build up the charge q already residing on the
sphere. And some additional work will be done moving in
additional increments of charge. So if we wanted to determine the
total amount of work that would be needed to assemble a total charge of
Q on the sphere then we need to integrate the expression above
We can think of the work done building up the charge as stored potential energy.
We can think of the work done building up the charge as stored potential energy.