If we move a charge q from a point a to a point b, then the
charge in potential energy, Uab, of the system is
is a small displacement along the
path we choose to take from a to b.
If we define the electric potential to be the potential energy per unit charge, so that V=U/q, then the difference in electric potential is given by,
With the definition of potential given in Eq. (2) and
taking the potential at infinity to be zero, we can
calculate the potential at each position from the
Electric potential due to a point charge
The electric field due to a point charge is
Uniform shell of charge
From the shell theorems we know that the electric
field inside a uniform shell of charge is zero.
In addition, if the electric field is zero the integral (2)
gives no contribution so
the electric potential is a constant. The electric
potential of a uniform shell of charge is
therefore a constant inside the shell of charge
and like a point charge centered at the origin
outside the shell of charge.
Inside and on the surface of conductors, the
electrical potential is a constant. This follows
as the electric field is zero inside and
parallel to the surface of conductors.
If we consider two parallel plates with uniform charge density , then the electric field between the plates has magnitude . Since this is a constant, the electric potential difference between the plates is simply E d where d is the distance between the plates. The electric potential is defined as the potential to move a small positive charge between two positions. In this case we imagine moving a small positive charge from the negative plate to the positive plate. This requires work and hence the electrical potential is positive. The electric potential of the positively charged plate is thus higher than the electric potential of the negatively charged plate.