Question

The figure shows two long parallel current carrying wires separated by a distance $5\text{m}$. The magnetic field at a point $\text{P}$, situated midway between the two wires will be

• A. $\frac{{\mu }_0}{4\pi }$
• B. $\frac{{\mu }_0}{\pi }$
• C. $\frac{{\mu }_0}{2\pi }$
• D. $4{\mu }_0\pi$

Answer 1

The correct option is C $\frac{{\mu }_0}{2\pi }$


Magnetic field at distance $r$ from a long current carrying wire is given by the formula,

$B=\frac{{\mu }_{0}i}{2\pi r}$

Now,

$\overrightarrow{B}$(due to wire $1$) at point $\text{P}$,

$\overrightarrow{B_1}=\frac{{\mu }_0{i}_1}{2\pi r}⨂$ (into the plane)

$i_1=5\text{A},r_1=2.5\text{m}$

$\overrightarrow{B_1}=\frac{5{\mu }_0}{2\pi \left(2.5\right)}⨂$

Assuming that the two wires lies in the $x-y$ plane, so

$\overrightarrow{B_1}=\frac{{\mu }_0}{\pi }(-\hat{k})$

Similarly, for wire $2$, magnetic field at point $\text{P}$,

$\overrightarrow{B_2}=\frac{2.5{\mu }_0}{5\pi }⨀($out of the plane or $\hat{k})$

$\overrightarrow{B_2}=\frac{{\mu }_0}{2\pi }\left(\hat{k}\right)$

So the net magnetic field at point $\text{P}$ is given by

$\overrightarrow{B_P}=\overrightarrow{B_1}+\overrightarrow{B_2}$

$\overrightarrow{B_P}=\frac{{\mu }_0}{\pi }(-\hat{k})+\frac{{\mu }_0}{2\pi }\hat{k}$

$\overrightarrow{B_P}=\frac{{\mu }_0}{2\pi }(-\hat{k})$

$\therefore \left|\overrightarrow{B_P}\right|=\frac{{\mu }_0}{2\pi }$

Therefore, option ($c$) is the correct answer.