Asteady current i flows through a wire of radius a. the current density in the wire varies with r as j = kr, where k is a constant and r is the distance from the axis of the wire. find expressions for the magnitudes of the magnetic field inside and outside the wire as a function of r. (hint: find the current through an ampèrian loop of radius r using ithru = j · da. use the following as necessary: μ0, i, a and r. do not substitute numerical values; use variables only.)a. b inside=? b. b outside=?

Explanation:

we can consider an element of radius r < a and thickness dr.  and Area of this element is

since current density is given

then , current through this element will be,

integrating on both sides between the appropriate limits,

Magnetic field can be found by using Ampere's law

for points inside the wire ( r<a)

now, consider a point at a distance 'r' from the center of wire. The appropriate Amperian loop is a circle of radius r.

by applying the Ampere's law, we can write

by symmetry will be of uniform magnitude on this loop and it's direction will be tangential to the loop.

Hence,

now using equation 1, putting the value of k,

B)

now, for points outside the wire ( r>a)

consider a point at a distance 'r' from the center of wire. The appropriate Amperian loop is a circle of radius l.

applying the Ampere's law

by symmetry will be of uniform magnitude on this loop and it's direction will be tangential to the loop. Hence

again using,equaiton 1,

continuous conducting path for current, power supply