Question

Two parallel long wires carry currents $i_1$ & $i_2$ ($i_1$> $i_2$) when the currents are in the same direction. The magnetic induction at a point midway between the two wires is $X$. If the direction of $i_2$ is reversed, the magnetic induction becomes $2X$, then $i_1$/ $i_2$ is:

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

Answer: (C)

$3$

At midway point when current is in same direction
$B=\frac{{\mu }_0{i}_1}{2\pi r}+\frac{{\mu }_0(-i_2)}{2\pi r}$
$=\frac{{\mu }_0}{2\pi r}(i_1-i_2)=X$

At midway point when current is in opposite direction
$B=\frac{{\mu }_0{i}_1}{2\pi r}+\frac{{\mu }_0\left(i_2\right)}{2\pi r}$
$=\frac{{\mu }_0}{2\pi r}(i_1+i_2)=2X$

So, $\frac{\frac{{\mu }_0}{2\pi r}(i_1+i_2)}{\frac{{\mu }_0}{2\pi r}(i_1-i_2)}=\frac{2X}{X}$
$\Rightarrow \frac{i_1+i_2}{i_1-i_2}=2$
$\Rightarrow \frac{i_1}{i_2}=3$