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Question

The current density across a cylindrical conductor of radius R varies according to the equation J=J0(1rR), where r is the distance from the axis. Thus, the current density is a maximum J0 at the axis r=0 and decreases linearly to zero at the surface r=R. Calculate the current, I in terms of J0 if the conductor's cross sectional area is A=πR2.

A
I=J0A3
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B
I=J0A6
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C
I=J0A9
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D
I=J0A12
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Solution

The correct option is A I=J0A3

Lets, take a cylindrical element of radius r and thickness dr.
Area of this element dA=2πrdr .

Then current through this area dA is,

di=J.dA=JdA

Substituting the value of J,

di=J0(1rR)2πrdr

Integrating on both sides,

Total current through the cross-section, i=di=R0J0(1rR)2πrdr

i=2πJ0[R22R33R]

i=J0πR23=J0A3

Hence, option (a) is the correct answer.

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