(a) The figure given below shows three straight long parallel conductors (1), (2) and (3) kept in x-y plane, carrying currents 2I, I and 3I…

Finding the magnitude and direction of:

Magnetic field due to conductor ($\displaystyle 2$) at a point on conductor ($\displaystyle 1$) is $\displaystyle \mathrm{B}_{1}=\frac{\mu_{0} \mathrm{I}}{2 \pi \mathrm{~d}}$ along positive z - axis.

Magnetic field due to conductor ($\displaystyle 3$) at a point on conductor ($\displaystyle 1$) is $\displaystyle \mathrm{B}_{2}=\frac{3 \mu_{0} \mathrm{I}}{4 \pi \mathrm{~d}}$ along negative z - axis.

Magnitude of net magnetic field is $\displaystyle \mathrm{B}_{2}-\mathrm{B}_{1}=\frac{3 \mu_{0} \mathrm{I}}{4 \pi \mathrm{d}}-\frac{\mu_{0} \mathrm{I}}{2 \pi \mathrm{d}}=\frac{\mu_{0} \mathrm{I}}{4 \pi \mathrm{d}}$ along negative $\displaystyle z-$ axis.

Direction will be along negative z axis/ into the plane of the paper.

Force per unit length on conductor ($\displaystyle 1$) due to conductor ($\displaystyle 2$) $\displaystyle =\frac{\mu_{0} \mathrm{I}_{1} \mathrm{I}_{2}}{2 \pi \mathrm{~d}}$

\[=\frac{\mu_{0} \mathrm{I}^{2}}{\pi \mathrm{~d}} \text { (attractive) } \]

Force per unit length on conductor ($\displaystyle 1$) due to conductor ($\displaystyle 3$) $\displaystyle =\frac{3 \mu_{0} \mathrm{I}^{2}}{2 \pi \mathrm{~d}}$ (repulsive)

Magnitude of net force per unit length on conductor ($\displaystyle 1$) due to ($\displaystyle 2$) & ($\displaystyle 3$) $\displaystyle =\frac{\mu_{0} \mathrm{I}^{2}}{2 \pi \mathrm{~d}}$

Direction of the net force per unit length will be repulsive or along positive $\displaystyle y$-axis.

Deriving expression for induced emf

Finding the effective value of current

\[\begin{aligned} \text { Induced emf } & =|\varepsilon|=\frac{\mathrm{d} \phi}{\mathrm{dt}} \\ & =\frac{\mathrm{d}}{\mathrm{dt}}(\overrightarrow{\mathrm{~B}} \cdot \overrightarrow{\mathrm{~A}}) \end{aligned} \]