Question

Hello! I really need help on this. All work shown would be awesome so I can understand the concepts and please write legibly! Thank you:)

(a) Consider a uniformly charged thin-walled right circular cylindrical shell having total charge Q, radius R, and length . Determine the electric field at a point a distance d from the right side of the cylinder as shown in the figure below. Suggestion: Use the following expression and treat the cylinder as a collection of ring charges. (Use any variable or symbol stated above along with the following as necessary: ke.) kex (x* + a2)3/2 (b) Consider now a solid cylinder with the same dimensions and carrying the same charge, uniformly distributed through its volume. Use the following expression to find the field it creates at the same point. (Use any variable or symbol stated above along with the following as necessary: ke.) (x2 + r2)1/2 Need Help? 1|Read

Answer 1

Consider a ring with thinness dx, and it has charge, Q dx/l dE = ke x/(x^2 + R^2)^3/2 Q dx/l i^- integral dt = integra_d^dt l ke Q x dx/h (x^2 + R^2)63/2 i^- = ke Q/2l integral_d^dt l (x^2 + R^2)^-3/2 x dx i^- = ke Q/2l (x^2 + R^2)^-1/2/(-1/2) Vertical-bar _d^dt l i^- E = ke Q/l [1/(d^2 + R^2)^1/2 - 1/[(dt l)^2 + R^2]^1/2]i^- (b) charge per unit area, sigma = Q dx/pi R^2 l dE = 2 pi ke Q dx/pi R^2 l (1 - x/(x^2 + R^2)^1/2) i^- E = integral_d^dt l 2 pi ke Q dx/pi R^2 l (1 - x/(x^2 + R^2)^1/2) i^- = 2 ke Q/R^2 l [x Vertical-bar _d^dt l - 1/2 (x^2 + R^2)^1/2/1/2 Vertical-bar _d^dt l] = 2 ke Q/R^2 l [d + l - d - [(dt l)^2 + R^2)^1/2 + (d^2 + R^2)^1/2] E = 2 ke Q/R^2 l [l + (d^2 + R^2)61/2 ((dt l)^2 + R^2)^1/2]