Question

A battery is connected between two points $A$ and $B$ on the circumference of a uniform conducting ring of radius $r$ and resistance $R$ . One of the arcs $AB$ of the ring subtends an angle $\theta $ at the centre. The value of the magnetic induction at the centre due to the current in the ring is

• A. proportional to $\left(\right.180^{\circ} -\theta \left.\right)$
• B. inversely proportional to $r$
• C. zero, only if $\left(\right.\theta =180^{\circ} \left.\right)$
• D. zero for all values of $\theta $

Answer 1

Answer: (D)

For a current flowing into a circular arc, the magnetic induction at the centre is
$B =\left(\frac{\mu_{0} i}{4 \pi}\right) \theta$
or $\text{B} \propto i \theta $
In the given problem, the total current is divided into two arcs
$i \propto \frac{1}{\text{resistance of arc}}$
$ \propto \frac{1}{\text{length of arc}}$
$ \propto \frac{1}{\text{angle subtended at centre } \left(\theta \right)}$



or $\text{i}\theta =\text{constant}$
i.e., magnetic field at centre due to arc $AB$ is equal and opposite to the magnetic field at centre due to arc $ACB$ . Or the net magnetic field at the centre is zero.