Question

Two long thin wires ABC and DEF are arranged as shown in the figure. They carry equal currents I. The magnitude of the magnetic field at O is

• A. Zero
• B. $\frac{{\mu }_{0}I}{4\pi a}$
• C. $\frac{{\mu }_{0}I}{2\pi a}$
• D. $\frac{{\mu }_{0}I}{2\sqrt{2}\pi a}$

Answer 1

The correct option is C $\frac{{\mu }_{0}I}{2\pi a}$

AB and DE do not give any magnetic field at O.

Magnetic field at distance $d$ due to finite wire is
$B=\frac{{\mu }_{o}I}{4\pi r}(\sin {\theta }_1+\sin {\theta }_2)$

Magnetic field at distance $d$ due to semi infinite wire is
$B=\frac{{\mu }_{o}I}{4\pi d}$ (as ${\theta }_1=0^{\circ }$ and ${\theta }_2={90}^{\circ }$)

Magnetic field at O due to BC is $B_{BC}=\frac{{\mu }_{O}I}{4\pi a}$ (Out of plane of paper)
Magnetic field at O due to EF is
$B_{EF}=\frac{{\mu }_{O}I}{4\pi a}$ (Out of plane of paper)

So the final $B$ is the sum of the BC and EF.
So, $B_{total}=B_{BC}+B_{EF}=\frac{{\mu }_{O}I}{2\pi a}$