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 potential V 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)

electric field in the wire

E(vector) = V / L ( x unit vector) ..................( in the direction of x)

b)

R = rho * L / A

R = rho * L / pie * (d / 2 )^2

R = 4 * rho * L / pie * d^2

the resistance of the wire is R = ( 4 * rho * L ) / pie * d^2

c)

electric current in the wire

I (vector ) = V / R (x unit vector )

then put the value of R

I ( vector ) = (pie * d^2 *V / 4 * rho * L )   x( unit vector in the direction of x)

d)

the current density in the wire

J ( vector ) = I ( vector ) / A

then put the value of I

J (vector) = [(pie * d^2 *V / 4 * rho * L )   x( unit vector) ]/ (pie * (d/2)^2 )

J (vector) = (V / rho * L) x( unit vector in the direction of x)

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