Q.5 State the formula for magnetic induction produced by a current in a circular arc of a wire. Hence find the magnetic induction at the centre of a current carrying circular loop.
Best answer
Answer: \(B = \dfrac{\mu_0 I \theta}{4\pi R}\) at the centre of an arc; for a full loop \(B = \dfrac{\mu_0 I}{2R}\).
For a circular arc of radius \(R\) carrying current \(I\) and subtending angle \(\theta\) (in radian) at its centre, the Biot–Savart law gives (each element \(dl\) is perpendicular to the radius, \(dl = R\,d\theta\)):
\[B = \int \frac{\mu_0}{4\pi}\frac{I\,dl}{R^2} = \frac{\mu_0 I}{4\pi R^2}(R\theta) = \frac{\mu_0 I\theta}{4\pi R}\]directed perpendicular to the plane of the arc.
Circular loop: a complete loop is an arc with \(\theta = 2\pi\):
\[B = \frac{\mu_0 I (2\pi)}{4\pi R} = \frac{\mu_0 I}{2R}\]For a coil of \(N\) turns, \(B = \dfrac{\mu_0 N I}{2R}\).
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