Question

If the current in a wire is given by j=kr,0<r<R, where R is the radius of the wire, the current of the wire is?
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The answer I get is : kr(pi) r^2.
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Muiltple Choice
A) 3 (pi) kr^3/2
B) kr^3/3
C) kr^2/2
D) kr^2
E) 2 (pi) kr^3/3

Answer 1

Take the circular element of the wire. The area of the small circular element is,

\(d A=2 \pi r d r\)

(1)

Therefore, the current through the wire is,

\(I=\int_{0}^{R} j d A\)

Substitute, the equation (1) in (2).

\(I=\int_{0}^{R} k r \cdot(2 \pi r d r)\)

Integrate the above equation.

\(I=\int_{0}^{R} k r(2 \pi r d r)\)

\(=2 \pi k \int_{0}^{R}\left(r^{2} d r\right)\)

\(=\frac{2 \pi k R^{3}}{3}\)

Hence, the option \([b]\) is correct.