Question

Magnetic field due to two current elements having same currents in same direction at a point $\text{P}$ in between them is zero. If $r$ is the distance of point $\text{P}$ from one of the conductors and $d$ is the distance between both the conductors, then the value of $d$ is

• A. $2r$
• B. $4r$
• C. $6r$
• D. $r$

Answer 1

The correct option is A $2r$

According to Biot-Savart Law,

Magnetic field due to any current element of length $dl$ is

$dB=\frac{{\mu }_0}{4\pi }\frac{idl\sin \theta }{r^2}$

As per question, the net magnetic field at $\text{P}$ is zero.

Therefore, magnetic fields at $\text{P}$ due to both the current elements are equal and in opposite direction.

So,

$d{B}_1=d{B}_2$

$\Rightarrow \frac{{\mu }_0}{4\pi }\frac{idl\sin \theta }{r^2}=\frac{{\mu }_0}{4\pi }\frac{idl\sin \theta }{(d-r{)}^2}$

$\Rightarrow r^2=(d-r{)}^2$

$\Rightarrow r^2=d^2+r^2-2dr$

$\Rightarrow d^2=2dr$

$\therefore d=2r$