First slide

Magnetic field due to a current carrying circular ring

Question

A circular coil of radius R carries an electric current. The magnetic field due to the coil at a point on the axis of the coil located at a distance r from the centre of the coil, such that r >> R, varies as

Moderate

A

$\frac{1}{r}$
B

$\frac{1}{r^{3}}$
C

$\frac{1}{r^{2}}$
D

$\frac{1}{r^{3 / 2}}$

Solution

The magnetic field due to a circular coil of radius Rat a point on the axis of the coil located at a distance r from the centre of the coil,
$B = \frac{μ_{0}}{4 π} \frac{2 {πiR}^{2}}{{(R^{2} + r^{2})}^{3 / 2}}$
Given, r >> R then we have, after neglecting R,
$B = \frac{μ_{0}}{4 π} \frac{2 {πiR}^{2}}{r^{3}}$
$\begin{array}{l} Also area A = {πR}^{2} \\ B = \frac{μ_{0}}{2 π} \frac{Ai}{r^{3}} \Rightarrow B \propto \frac{1}{r^{3}} \end{array}$

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