Question

1. What is the potential at the center of a charged circular loop with a total charge of Q and radius R?
2. A point charge q is moved a distance d parallel to the x-axis in a region of space where there is an electric field of magnitude E which is parallel to the y-axis. What is the change in the potential energy of the charge?
3. A collection of point charges exists in a region of space. Is it possible to arrange the charges so that the potential at a point P is equal to zero?
4. If the potential is constant in a certain region of space, what can you say about the electric field there?

Answer 1

Consider a small length "dl" of the complete loop having small charge "dq"

Q = total charge on the loop

R = radius of the loop

L = length of the loop = circumference = 2R

small charge on the small length is given as

dq = (Q/(L)) dl

dq = (Q/(2R)) dl

Small electric potential at the center due to small length is given as

dV = k dq/R

dV = k (Q/(2R)) dl/R

Total electric potential is given as

V = k (Q/(2R)) /R dl

V = k (Q/(2R)) (2R /R )

V = kQ/R

Since the direction of motion of the charge is perpendicular to the electric field, and we know that along the perpendicular direction of electric field, electric potential does not change., hence change in potential energy of the charge is zero.