Question

Equal current $i$ is flowing in two parallel infinite long conductors at distant $2l$, in mutually opposite directions as shown in the figure. The magnetic induction at a point $p$ equidistant $r$ from both conductors will be :

• A. $\frac{{\mu }_{0}il}{\pi r^2}$
• B. $\frac{{\mu }_{0}i{r}^2}{\pi r}$
• C. $\frac{{\mu }_{0}i{l}^2}{\pi r^2}$
• D. $\frac{{\mu }_{0}il}{\pi r}$

Answer 1

Answer: (A)

$\frac{{\mu }_{0}il}{\pi r^2}$

According to Right hand thumb rule, the direction of magnetic induction at point P due to the wires is shown in the figure which we resolve as horizontal components and vertical components. The vertical components cancel each other and thus the total magnetic induction at P is in horizontal direction.

Magnetic induction due to infinite wire at a perpendicular distance $d$ is given by $B_w=\frac{{\mu }_{o}i}{2\pi d}$

$\therefore$ Magnetic induction due to each wire $B=\frac{{\mu }_{o}i}{2\pi r}$ .........(1)

From geometry $sin\theta =\frac{l}{r}$

Thus total magnetic induction at point P $B_T=2Bsin\theta$

$\therefore B_T=2\times \frac{{\mu }_{o}i}{2\pi r}\times \frac{l}{r}$ $⟹B_T=\frac{{\mu }_{o}il}{\pi r^2}$