Question

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.

Hint:

Start with the general arc formula \(B = \frac{\mu_0 I \theta}{4\pi r}\). For a full circle, the angle \(\theta\) is \(2\pi\). For a semi-circle, \(\theta\) is \(\pi\). This makes it easy to find the field for any part of a circle.

Answer 1

According to the Biot-Savart law, the magnetic induction (\(B\)) at the centre of a circular arc of wire with radius \(r\), carrying a current \(I\), and subtending an angle \(\theta\) (in radians) at the centre is given by: \[ B = \frac{\mu_0 I \theta}{4\pi r} \] where \(\mu_0\) is the permeability of free space.
Magnetic Induction at the Centre of a Circular Loop:
To find the magnetic induction at the centre of a full circular loop, we consider the arc to be a complete circle.
For a complete circle, the angle \(\theta\) subtended at the centre is \(2\pi\) radians.
Substituting \(\theta = 2\pi\) into the formula for a circular arc: \[ B_{loop} = \frac{\mu_0 I (2\pi)}{4\pi r} \] \[ B_{loop} = \frac{\mu_0 I}{2r} \] This is the formula for the magnetic induction at the centre of a current-carrying circular loop.