First slide

Magnetic field due to a current carrying circular ring

Question

In a bent wire shown in figure, a current I is passed. Find the value of B at the common centre.

Moderate

A

$\frac{μ_{0} I (b - a) (2 π - θ)}{4 πab}$
B

$\frac{μ_{0} la (2 π - θ)}{4 πb}$
C

$\frac{μ_{0} la (2 π - θ)}{4 π a}$
D

$\frac{μ_{0} lab θ}{4 π}$

Solution

The straight segments make no contribution to the field at the centre. By Biot-Savart's law, the field at the centre due to curved segments is
$\begin{array}{l} ΔB = \frac{μ_{0} I}{4 π} [\frac{a (2 π - θ)}{a^{2}} - \frac{b (2 π - θ)}{b^{2}}] \\ = \frac{μ_{0} I}{4 π} (2 π - θ) [\frac{1}{a} - \frac{1}{b}] = \frac{μ_{0} I (2 π - θ) (b - a)}{4 πab} \end{array}$

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