Derive the electric potential for a discrete or continuous d

Derive the electric potential for a discrete or continuous distribution at a specific point in space. You should know how to derive the electric potential for the case of the uniform line, ring, and disk charges, and you should know how to describe the electric potential due to a dipole.

Solution

Electric Potential due to a Point Charge

Electric potential is a scalar quantity. It is the electric potential energy per unit charge and as such is a characteristic of the electric influence at that point in space. Since it is a scalar quantity, the potential from multiple point charges is just the sum of the point charge potentials of the individual charges and can be extended to calculate the potential from a continuous charge distribution.

Consider the electric potential due to a point charge q, As we move from point A, at distance r_A from the charge q, to point B, at distance r_B from the charge q, the change in electric potential is

delta( V_ba) = V_b-V_a = - integrating from a to b (E.ds)

E.ds = [ K q/r² ] r.ds

r means a unit vector which points a radial direction.

r.ds = dr

Only the radial distance r determines the work done or potential. We can move through any angle we like and, as long as the radial distance remains constant , no work is done or there is no change in the electric potential.

delta( V_ba) = V_b-V_a = - integrating from r_a to r_b (E.dr) = - integrating from r_a to r_b [[ K q/r² ] r.dr

delta( V_ba) = V_b-V_a = -Kq integration from r_a to r_b [1/r²] dr = -Kq  integration from r_a to r_b [r^-2]dr

delta( V_ba) = V_b-V_a= -Kq [(-1) r^-1] from r_a to r_b

delta( V_ba) = V_b-V_a= Kq [1/r] from r_a to r_b

delta( V_ba) = V_b-V_a= Kq [1/r_b-1/ r_a ]

This is the change in electric potential due to a point charge as we move from r_a to r_b.

The change in electric potential energy as we move from charge q^\' from radius r_a to r_b due to a point charge q.

delta U_ba =Kq\'q  [1/r_b-1/ r_a ]

As with gravitational potential energy or the electric potential energy relative to some reference point. We will choose that reference point to be infinity. that is, r_a= infinity

Then at some radius r , the electric potential is

V = Kq/r

If we bring in a charge q\', a distance r, from charge q, (or the \"system\") will have electric potential energy of

U = Kq\'q/r