Question

In the figure shown, two long wires $w_1$ and $w_2$, each carrying current $I$ are placed parallel to each other and parallel to $z$- axis. The direction of current in $w_1$ and $w_2$ is perpendicularly outward and inward to the plane respectively. The $\overrightarrow{B}$ at point $\text{P}$ will be given as:

• A. $\overrightarrow{B}=\frac{2{\mu }_{0}I}{5\pi a}\hat{i}+\frac{{\mu }_{0}I}{5\pi a}\hat{j}$
• B. $\overrightarrow{B}=\frac{{\mu }_{0}I}{2\pi a}\hat{i}-\frac{{\mu }_{0}I}{5\pi a}\hat{j}$
• C. $\overrightarrow{B}=\frac{2{\mu }_{0}I}{5\pi a}\hat{j}$
• D. $\overrightarrow{B}=\frac{{\mu }_{0}I}{5\pi a}\hat{i}-\frac{{\mu }_{0}I}{5\pi a}\hat{j}$

Answer 1

The correct option is A $\overrightarrow{B}=\frac{2{\mu }_{0}I}{5\pi a}\hat{i}+\frac{{\mu }_{0}I}{5\pi a}\hat{j}$

By using right- hand thumb rule we can find the direction of $\overrightarrow{B}$ at point $\text{P}$ due to both wires.

From geometry,

$\sin \theta =\frac{a}{\sqrt{5}a}=\frac{1}{\sqrt{5}}$ and $\cos \theta =\frac{2a}{\sqrt{5}a}=\frac{2}{\sqrt{5}}$

The magnetic field due to infinite wire is given by

$B=\frac{{\mu }_{0}I}{2\pi d}$

For wire $w_2$, perpendicular distance from point $\text{P}$ is $d_2=a$

And for wire $w_1$, perpendicular distance from point $\text{P}$ is
$d_1=\sqrt{4a^2+a^2}=\sqrt{5}a$

So,
$B_1=\frac{{\mu }_0{I}_1}{2\pi d_1}=\frac{{\mu }_{0}I}{2\sqrt{5}\pi a}$

And, $B_2=\frac{{\mu }_0{I}_2}{2\pi d_2}=\frac{{\mu }_{0}I}{2\pi a}$

So, the net magnetic field at point $\text{P}$ due to both wires will be

${\overrightarrow{B}}_{net}={\overrightarrow{B}}_1+{\overrightarrow{B}}_2$

${\overrightarrow{B}}_{net}=-B_1\sin \theta \hat{i}+B_1\cos \theta \hat{j}+B_2\hat{i}$

putting the value of given data,

${\overrightarrow{B}}_{net}=-\frac{{\mu }_{0}I}{2\sqrt{5}\pi a}\frac{1}{\sqrt{5}}\hat{i}+\frac{{\mu }_{0}I}{2\sqrt{5}\pi a}\frac{2}{\sqrt{5}}\hat{j}+\frac{{\mu }_{0}I}{2\pi a}\hat{i}$

$\therefore {\overrightarrow{B}}_{net}=\frac{2{\mu }_{0}I}{5\pi a}\hat{i}+\frac{{\mu }_{0}I}{5\pi a}\hat{j}$

Hence, option $\left(a\right)$ is correct.