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Question

A long cylindrical conductor of radius R carries current i as shown below. The current density J is a function of radius r as J=br where b is a constant. The magnetic field at a distance r1 (r1<R) is:


A
μ0br212
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B
Zero
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C
μ0br312
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D
μ0br213
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Solution

The correct option is D μ0br213

Let us consider an elemental ring of thickness dr at a distance r from the centre.

So area of the element will be,
dA=2πrdr

Now the current flowing through this element,

di=JdA=br(2πrdr)

di=2πbr2dr

Considering the direction of field lines as per current direction , the enclosed current within the amperian loop (r1<R) will be

Ien=di=r1r=02πbr2dr

Ien=2πbr313

Applying ampere's circuital law for loop;

B.dl=μ0Ien

B(2πr1)=μ0(2πbr313)

B=μ0br213

Hence, option (c) is correct.

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