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.