Magnetics 2 Question 16
16. 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 $\operatorname{arcs} A B$ 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
$(1995,2 M)$
(a) proportional to $\left(180^{\circ}-\theta\right)$
(b) inversely proportional to $r$
(c) zero, only if $\left(\theta=180^{\circ}\right)$
(d) zero for all values of $\theta$
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Answer:
Correct Answer: 16. (c)
Solution:
- For a current flowing into a circular arc, the magnetic induction at the centre is
$$ B=\frac{\mu _0 i}{4 \pi r} \theta \text { or } B \propto i \theta $$
In the given problem, the total current is divided into two arcs
$$ \begin{aligned} i & \propto \frac{1}{\text { resistance of arc }} \propto \frac{1}{\text { length of arc }} \\ & \propto \frac{1}{\text { angle subtended at centre }(\theta)} \\ i \theta & =\text { constant } \end{aligned} $$
or
i.e. magnetic field at centre due to $\operatorname{arc} A B$ is equal and opposite to the magnetic field at centre due to $\operatorname{arc} A C B$. Or the net magnetic field at centre is zero.
