Question

A V shape bent wire is carrying current $I$ and ends of wire extends to infinity. If the magnetic field at $P$ can be written as $K\tan \left(\frac{\alpha }{2}\right)$, then $K$ is

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

Answer 1

The correct option is B $\frac{{\mu }_{0}I}{2\pi d}$

Magnetic field due to any one segment:

$B=\frac{{\mu }_{0}i}{4\pi \left(d\sin \alpha \right)}(\cos 0^o+\cos (180-\alpha ))$

$=\frac{{\mu }_{0}i}{4\pi \left(d\sin \alpha \right)}(1-\cos \alpha )=\frac{{\mu }_{0}I}{4\pi d}\tan \frac{\alpha }{2}$

Resultant field will be

$B_{net}=2B=\frac{{\mu }_{0}I}{2\pi d}\tan \frac{\alpha }{2}\Rightarrow K=\frac{{\mu }_{0}I}{2\pi d}$