Question

Suppose a particle is constrained to move along the x-axis. There are two rings of contin- uous charge distributions (total charge each of Q) centered on the x-axis and on a plane perpendicular to it. One ring is located at x L and the other at x =-L. (a) Obtain an expression for the potential due to the ring charges as a function of z for all x-values in between a E(-L, L). Hint: you should be integrating something. b) Show that V() has an extrema at a 0 (in fact, it is a minimum). Then compute the potential to find its value there.

Answer 1

Consider a location at distance x from origin. Consider an element dq on both ring so, potential at point x. V = 1/4 pi x_0 dq/squareroot R^2 + (L + x)^2 + 1/4 pi x_0 dq/squareroot R^2 + (L - x)^2 For entire ring V = Q/4 pi x_0 [1/squareroot R^2 + (L + x)^2 + 1/squareroot R^2 + (L - x)^2] Differenting V with respect to x_1 dV/dx = Q/4 pi x_0 [-1/2 2 (L + x)/[R^2 + (L + x)^2]^3/2 + 1/2 - 2 (L - x)/[R^2 + (L - x)^2]^3/2 At x = 0 dV/dx = 0 So, V (x) has extreme at x = 0 At x = 0 V = Q/2 pi x_0 1/squareroot R^2 + L^2