An infinitely long hollow conducting cylinder with inner radius R / 2 and outer radius R carries a uniform current density along its length. The magnitude of the magnetic field, B as a function of the

Question

An infinitely long hollow conducting cylinder with inner radius R / 2 and outer radius R carries a uniform current density along its length. The magnitude of the magnetic field, B as a function of the radial distance r from the axis, is best represented by

TestHub · Accepted Answer

The required cylinder is shown in figure

Let $j =$ current density, then

For, $r < \frac{R}{2}$

$B = 0$

For, $\frac{R}{2} \leq r \leq R$

$B \times 2 πr = μ_{0} jπ (r^{2} - \frac{R^{2}}{4})$

$B = \frac{μ_{0} j}{2} (r - \frac{R^{2}}{4 r})$

For, $r > R$

$B \times 2 πr = μ_{0} jπ (R^{2} - \frac{R^{2}}{4})$

$B = \frac{μ_{0} j}{2} \cdot \frac{2 R^{2}}{4 r}$

$\Rightarrow B \propto 1 / r$ [this is an equation for rectangular hyperbola.]

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