Question

Two identical copper wires $W_{1}and{W}_2$ placed in parallel as shown in the figure, carry currents I and 2I, respectively. In opposite directions. If the two wires are separated by a distance of 4r, then the magnitude of the magnetic field $\overrightarrow{B}$ between the wires at a distance r from $W_1$ is

• A. $\frac{{\mu }_0^2{I}^2}{2\pi r^2}$
• B. $\frac{6{\mu }_{0}I}{5\pi r}$
• C. $\frac{{\mu }_{0}I}{6\pi r}$
• D. $\frac{5{\mu }_{0}I}{6\pi r}$

Answer 1

The correct option is D $\frac{5{\mu }_{0}I}{6\pi r}$

Ans : (d)



Magnetic flux density $\left(\overrightarrow{B}\right)$ at r distance due to infinite line carrying current I is $\left|\overrightarrow{B}\right|=\frac{{\mu }_{0}I}{2\pi r}.$

• $\overrightarrow{B}$ at r distance due to $W_1$ wire

$=|{\overrightarrow{B}}_1|=\frac{{\mu }_{0}I}{2\pi r}$ ... (i)

• $\overrightarrow{B}$ at 3r distance due to $W_2$ wire

$=\left|{\overrightarrow{B}}_2\right|=\frac{{\mu }_0\left(2I\right)}{2\pi \left(3r\right)}$ ... (ii)

From right hand thumb rule, $\overrightarrow{B}$ due to both lines add in between conductors.
So, $\left|\overrightarrow{B}\right|=\left|{\overrightarrow{B}}_1\right|+\left|{\overrightarrow{B}}_2\right|$

$\therefore \left|\overrightarrow{B}\right|=\frac{{\mu }_{0}I}{2\pi r}+\frac{2{\mu }_{0}I}{6\pi r}=\frac{5{\mu }_{0}I}{6\pi r}$