Question

Two long straight horizontal parallel wires one above the other are separated by a distance $2a$. If the wires carry equal currents in opposite directions, the magnitude of the magnetic induction in the plane of the wires at a distance $a$ above the upper wire is

• A. $\frac{{\mu }_{0}i}{2\pi a}$
• B. $\frac{{\mu }_{0}i}{2\pi a}+\frac{{\mu }_{0}i}{4\pi a}$
• C. $\frac{{\mu }_{0}i}{2\pi a}-\frac{{\mu }_{0}i}{4\pi a}$
• D. $\frac{{\mu }_{0}i}{3\pi a}$

Answer 1

Answer: (D)

$\frac{{\mu }_{0}i}{3\pi a}$

So, at a height from top wire $B=\frac{{\mu }_{0}i}{2\pi a}$
and from bottom wire $B=\frac{{\mu }_0(-i)}{2\pi \left(3a\right)}$
So, $B_{total}=\frac{{\mu }_{0}i}{2\pi a}+\frac{{\mu }_0(-i)}{2\pi \left(3a\right)}$
$=\frac{{\mu }_{0}i}{2\pi r}\left(1-\frac{1}{3}\right)$
$=\frac{{\mu }_{0}i}{2\pi a}\times \frac{2}{3}$
$=\frac{{\mu }_{0}i}{3\pi a}$