Question

Two wires $A$ and $B$ are of lengths $40cm$ and $30cm$ . $A$ 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 $=40cm$
It is bent into a circle, so
$\therefore 2\pi r=40$
$r=\frac{40}{2\pi }$
Magnetic induction at centre $=\frac{{\mu }_0{i}_1}{2r}$
For wire B Length $=30cm$ 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, $\frac{{\mu }_0{i}_1}{2r}=\frac{{\mu }_0{i}_1\theta }{4\pi r}$
$\Rightarrow \frac{i_1}{i_2}=\frac{\theta }{2\pi }=\frac{30\times 2\pi }{2\pi \times 40}$
$\Rightarrow \frac{i_1}{i_2}=\frac{3}{4}$