First slide

Magnetic field due to a current carrying circular ring

Question

In figure, infinite conducting rings each having current i in the direction shown are placed concentrically in the same plane as shown in the figure. The radius of rings are $r, 2 r, 2^{2} r, 2^{3} r . .... \infty$ The magnetic field at the centre of rings will be

Difficult

A

Zero
B

$\frac{μ_{0} i}{r}$
C

$\frac{μ_{0} i}{2 r}$
D

$\frac{μ_{0} i}{3 r}$

Solution

$\begin{array}{l} B = \frac{μ_{0} i}{2 r} - \frac{μ_{0} i}{2 (2 r)} + \frac{μ_{0} i}{2 (2^{2} r)} . ..... \infty \\ B = \frac{μ_{0} i}{2 r} [1 - \frac{1}{2} + \frac{1}{2^{2}} + \frac{1}{2^{3}} + . ... \infty] \\ B = \frac{μ_{0} i}{2 r} [\frac{1}{1 - (- \frac{1}{2})}] \\ B = \frac{μ_{0} i}{3 r} \end{array}$

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