Question

Two coils P and Q are kept near each other. When no current flows through coil P and current increases in coil Q at the rate $10\,{\text{A/s}}$, the emf in coil P is $15\,{\text{mV}}$. When coil Q carries no current and current of $1.8\,{\text{A}}$ flows through coil P, the magnetic flux linked with the coil Q is:
A. $1.4\,{\text{mWb}}$
B. $2.2\,{\text{mWb}}$
C. $2.7\,{\text{mWb}}$
D. $2.9\,{\text{mWb}}$

Answer 1

Use the formula for het emf induced in the secondary coil in terms of the mutual inductance and rate of change of current in the primary coil. Also use the formula for the magnetic flux linked with the coil in terms of the mutual inductance and current in the coil. First calculate the mutual inductance for the coils P and Q. Then calculate the magnetic flux linked with the coil Q.

Formulae used:
The formula for emf $e$ induced in secondary coil is given by
$e = M\dfrac{{di}}{{dt}}$ …… (1)
Here, $M$ is the mutual inductance and $\dfrac{{di}}{{dt}}$ is the rate of change of electric current in the primary coil.
The magnetic flux $\phi$ linked with the coil is given by
$\phi = Mi$ …… (2)
Here, $M$ is the mutual inductance between the coils and $i$ is the current in the coil.

Complete step by step answer:
We have given that two coils P and Q are placed near each other.When no current flows through the coil P, the rate of the current increasing in the coil Q is $10\,{\text{A/s}}$ and the emf induced in the coil is $15\,{\text{mV}}$.
$\dfrac{{d{i_Q}}}{{dt}} = 10\,{\text{A/s}}$
${e_P} = 15\,{\text{mV}}$
When there is no current in the coil Q, the current in the coil P is $1.8\,{\text{A}}$.
${i_P} = 1.8\,{\text{A}}$
We have asked to calculate the magnetic flux linked with the coil Q.

Let us first calculate the mutual inductance between the two coils P and Q.Rewrite equation (1) for the emf induced in the coil P.
${e_P} = M\dfrac{{d{i_Q}}}{{dt}}$
Rearrange the above equation for the mutual inductance $M$.
$M = \dfrac{{{e_P}}}{{\dfrac{{d{i_Q}}}{{dt}}}}$
Substitute $15\,{\text{mV}}$ for ${e_P}$ and $10\,{\text{A/s}}$ for $\dfrac{{d{i_Q}}}{{dt}}$ in the above equation.
$M = \dfrac{{15\,{\text{mV}}}}{{10\,{\text{A/s}}}}$
$\Rightarrow M = 1.5\,{\text{mH}}$
Hence, the mutual inductance between the coils P and Q is $1.5\,{\text{mH}}$.

Let now calculate the magnetic flux linked with the coil Q.Rewrite equation (2) for the magnetic flux linked with the coil Q.
${\phi _Q} = M{i_Q}$
Substitute $1.5\,{\text{mH}}$ for $M$ and $1.8\,{\text{A}}$ for ${i_Q}$ in the above equation.
${\phi _Q} = \left( {1.5\,{\text{mH}}} \right)\left( {1.8\,{\text{A}}} \right)$
$\therefore {\phi _Q} = 2.7\,{\text{mWb}}$
Therefore, the magnetic flux linked with the coil Q is $2.7\,{\text{mWb}}$.

Hence, the correct option is C.

Note: The students should not get confused between the rate of change of electric current and emf associated with the primary and secondary coils. In the present question, the primary coil is P and the secondary coil is Q. If the one gets confused between these values then we will end with the incorrect value of the magnetic flux linked with the coil Q.