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Question

The current density in a wire of radius a varies with radial distance r as J=kr2 , where k is a constant .

A
The total current passing through the cross section of the wire is I=πka42
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B
The total current passing through the cross section of the wire is I=3πka42
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C
The magnetic field at a distance r>a is B=μ0ka44r
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D
The magnetic field at a distance r<a is B=μ0kr34
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Solution

The correct option is D The magnetic field at a distance r<a is B=μ0kr34
Current through the wire of area da is
dI=Jda
dI=kr2(2πrdr)
I=2πka0r3dr


I=12πka4
Further, field for r>a is
B=μ0I2πr=μ0ka44r
and field for r<a is
B=μ0(πkr42)2πr=14μ0kr3

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