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Q.

An infinitely long, solid conducting cylindrical rod of radius $R$ is carrying a current distributed uniformly throughout the cross-section. What is the ratio of magnetic fields at perpendicular distances $\frac{R}{2}$ and $2 R$ from the axis of cylinder?

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Detailed Solution

The point at distance $\frac{R}{2}$ from the axis is inside the cylinder.
$B_{i} = \frac{μ_{0} I}{2 π R^{2}} r = \frac{μ_{0} I}{2 π R^{2}} \times \frac{R}{2} = \frac{μ_{0} I}{4 π R} ........ (1)$
The point at distance $2 R$ from the axis is outside the cylinder.
$B_{O} = \frac{μ_{0} I}{2 π (2 R)} = \frac{μ_{0} I}{4 π R} ......... (2)$
$\Rightarrow \frac{B_{i}}{B_{O}} = 1$

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