A straight, cylindrical wire lying along the x axis has a length L and a diamete

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 . 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, , and physical constants, derive expressions for the following. (Use the following as necessary: L, d, V, , and .)

(a) the electric field in the wire

(b) the resistance of the wire
R =  

(c) the electric current in the wire

(d) the current density in the wire

Explanation / Answer

a)   E = V/L x^

b) R = L/A = L/(d/2)^2 = 4L/d^2

c) I = V/R x^ = V/(4L/d^2) x^ = d^2*V/4L x^

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

e) J = (V/L x^) = V/L x^ = E

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