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Q.

The current density in a wire of radius ‘a’ varies with r according to $j = k r,$ where k is a constant and r is the distance from the axis of wire the magnetic field at a point distant r from the axis $(r < a)$ is $\frac{μ_{0} k r^{2}}{x} .$ then value of x is __________________.

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answer is 3.

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Detailed Solution

Question Image
$d I = j d A = k r \cdot 2 π r d r$
$I = 2 π k \int_{0}^{r} r^{2} d r = \frac{2 π k r^{2}}{3}$
$B \cdot 2 π r = μ_{0} I = \frac{μ_{0} π λ R^{4}}{2}$
$B = \frac{μ_{0} π λ R^{4}}{4 r}$
$r < R$
$B \cdot 2 π r = μ_{0} \cdot 2 π λ \int_{0}^{r} r^{3} d r = μ_{0} \cdot 2 π λ \frac{r^{4}}{4}$
$B = \frac{μ_{0} λ r^{3}}{4}$

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