Mathematics

Problem

A long, solid cylindrical conductor of radius

R

carries a steady current whose volume current density varies with radial distance according to

J(r)=J_{0}\dfrac{r}{R}

, where

J_{0}

is a constant. a. (i) Obtain an expression for the total current

I

carried by the conductor in terms of

J_{0}

and

R

. b. (ii) Derive an expression for the magnitude of the magnetic field

B(r)

at a distance

r

from the axis for points inside the conductor

(r<R)

. c. (iii) Derive

B(r)

for points outside the conductor

(r\ge R)

. d. (iv) On a single set of qualitative axes, sketch the variation of

B

with

r

from

r=0

r=3R

. Label the key features of the graph.