Question

A straight, cylindrical wire lying along the x axis has a length L and a diameter d. It is made of a material described by Ohm's law with a resistivity rho. Assume potentialV is maintained at the left end of the wire at x = 0. Also assume the potential is zero at x = L. In terms of L, d, V, rho, and physical constants, derive expressions for the following. (Use the following as necessary: L, d, V, rho, and π.)(a) the electric field in the wiremagnitude direction ---Select--- +i -i +j -j +k -k(b) the resistance of the wireR = (c) the electric current in the wiremagnitude direction ---Select--- +i -i +j -j +k -k(d) the current density in the wiremagnitude direction ---Select--- +i -i +j -j +k -k(e) Show that E with arrow = rhoJ with arrow. (Do this on paper. Your instructor may ask you to turn in this work.)

Answer 1

a) E = V/L x^

b) R = ρL/A = ρL/π(d/2)^2 = 4ρL/πd^2

c) I = V/R x^ = V/(4ρL/πd^2) x^ = πd^2*V/4ρL x^

d) J = I/A = [πd^2*V/4ρL x^]/π(d/2)^2 = V/ρL x^

e) ρJ = ρ(V/ρL x^) = V/L x^ = E