Question

Two infinitely large sheets $\text{A}$ and $\text{B}$ carrying current in same direction per unit length in $\text{A}$ and $\text{B}$ are $5\text{A/m}$ and $10\text{A/m}$ respectively. Then find the ratio of magnetic field at $p$ and $q$.

• A. $3$
• B. $\frac{1}{9}$
• C. $9$
• D. $\frac{1}{3}$

Answer 1

The correct option is A $3$


Magnetic field at $p$ due to sheet $\text{A}$ is,

${\overrightarrow{B}}_{Ap}=\frac{{\mu }_0{\lambda }_A}{2}(\leftarrow )$

With the help of right- hand screw rule, we can find the direction of magnetic field.

Magnetic field at $p$ due to sheet $\text{B}$ is,

${\overrightarrow{B}}_{Bp}=\frac{{\mu }_0{\lambda }_B}{2}(\leftarrow )$

So, net magnetic field at point $p$,

$|{\overrightarrow{B}}_p|=|{\overrightarrow{B}}_{Ap}+{\overrightarrow{B}}_{Bp}|$

$|{\overrightarrow{B}}_p|=\frac{{\mu }_o{\lambda }_B}{2}+\frac{{\mu }_o{\lambda }_A}{2}=\frac{{\mu }_o}{2}({\lambda }_B+{\lambda }_A)$

Similarly, magnetic field at $q$ due to sheet $A$ is,

${\overrightarrow{B}}_{Aq}=\frac{{\mu }_0{\lambda }_A}{2}(\to )$

Magnetic field at $q$ due to sheet $B$ is,

${\overrightarrow{B}}_{Bq}=\frac{{\mu }_0{\lambda }_B}{2}(\leftarrow )$

So, net magnetic field at point $q$,

$|{\overrightarrow{B}}_q|=|{\overrightarrow{B}}_{Aq}+{\overrightarrow{B}}_{Bq}|$

$|{\overrightarrow{B}}_q|=\frac{{\mu }_o{\lambda }_B}{2}-\frac{{\mu }_o{\lambda }_A}{2}=\frac{{\mu }_o}{2}({\lambda }_B-{\lambda }_A)$

Taking ratio of magnitude of both magnetic fields

$\frac{|{\overrightarrow{B}}_p|}{|{\overrightarrow{B}}_q|}=\frac{\frac{{\mu }_0}{2}({\lambda }_A+{\lambda }_B)}{\frac{{\mu }_0}{2}({\lambda }_B-{\lambda }_A)}$

$\therefore \frac{|{\overrightarrow{B}}_p|}{|{\overrightarrow{B}}_q|}=\frac{{\lambda }_A+{\lambda }_B}{{\lambda }_B-{\lambda }_A}=\frac{10+5}{10-5}=3$

Hence, option $\left(A\right)$ is the correct answer.