Question

In Fig. $\left(a\right)$ and Fig. $\left(b\right)$, two air-cored solenoids $P$ and $Q$ have been shown.

They are placed near each other. In Fig. $\left(a\right)$, when $I_P$ , the current in $P$, changes at the rate of $5\text{A/s}$, an emf of $2\text{mV}$ is induced in $Q$. The current in $P$ is then switched off, and a current changing at $2\text{A/s}$ is fed through $Q$ as shown in diagram. What emf will be induced in solenoid $P$?

• A. $5\times {10}^{-3}\text{V}$
• B. $2\times {10}^{-3}\text{V}$
• C. $8\times {10}^{-4}\text{V}$
• D. $8\times {10}^{-2}\text{V}$

Answer 1

The correct option is C $8\times {10}^{-4}\text{V}$

$\Rightarrow$Mutual inductance $\left(M\right)$ depends on geometric shape and relative orientation of two coils. Hence for both cases the $M$ will be same.
$\Rightarrow$Between solenoid $P$ and $Q$ if current in $P$ is varrying
$\left|\frac{d{i}_1}{dt}\right|=5\text{A/s}$
$\Rightarrow {\epsilon }_1=M\left|\frac{d{i}_1}{dt}\right|$
$2\times {10}^{-3}=M\times 5$
$\therefore M=0.4\times {10}^{-3}...\left(i\right)$
$\Rightarrow$Between solenoid $P$ and $Q$ if current in $Q$ is varrying:
$\left|\frac{d{i}_2}{dt}\right|=2\text{A/s}$
$\Rightarrow {\epsilon }_2=M\left|\frac{d{i}_2}{dt}\right|$
Putting the value of $M$ from Eq $\left(i\right)$,
${\epsilon }_2=0.4\times {10}^{-3}\times 2\text{V}$
$\therefore {\epsilon }_2=8\times {10}^{-4}\text{V}$
Hence ${\epsilon }_2$ is the emf induced in solenoid $P$.