Question

Two wires $A$ and $B$ are of lengths $ 40\,cm $ and $ 30\,cm $ . $A$ is bent into a circle of radius $r$ and $B$ into an arc of radius $r$. A current $i_{1}$ is passed through $A$ and $I_2$ through $B$. To have the same magnetic inductions at the centre, the ratio of $I_{1}:I_{2}$ is

• A. $ 3:4 $
• B. $ 3:5 $
• C. $ 2:3 $
• D. $ 4:3 $

Answer 1

Answer: (A)

For wire A Length $=40\, cm$
It is bent into a circle, so
$\therefore 2 \pi r=40$
$r=\frac{40}{2 \pi}$
Magnetic induction at centre $=\frac{\mu_{0} i_{1}}{2 r}$
For wire B Length $=30 cm$ It is bent into an arc, so
$\therefore \theta r=30$
Magnetic induction at centre due to circular arc
$=\frac{\mu_{0} i_{2} \theta}{4 \pi r}$
Since, $\frac{\mu_{0} i_{1}}{2 r}=\frac{\mu_{0} i_{1} \theta}{4 \pi r}$
$\Rightarrow \frac{i_{1}}{i_{2}}=\frac{\theta}{2 \pi}=\frac{30 \times 2 \pi}{2 \pi \times 40}$
$\Rightarrow \frac{i_{1}}{i_{2}}=\frac{3}{4}$