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There are two coils A and B separated by some distance. If a current of $2 \mathrm{~A}$ flows through $\mathrm{A}$, a magnetic flux of $10^{-2}$ Wb passes through $B$ (no current through $B$ ). If no current passes through $A$ and a current of 1A passes through $B$, what is the flux through $\mathrm{A}$ ?
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Verified Answer
Let $\mathrm{A}$ current $\mathrm{I}_1$ is passing through the coil $\mathrm{A}$ having mutual inductance $\mathrm{M}_{21}$ with respect to coil $\mathrm{B}$ is,
$$
\mathrm{N}_2 \phi_2=\mathrm{M}_{21} \mathrm{I}_1
$$
where, $\mathrm{M}_{21}$ is called the mutual inductance of coil A with respect to coil $\mathrm{B}$ and $\mathrm{M}_{21}=\mathrm{M}_{12}$ and $\mathrm{M}_{12}$ is called the mutual inductance of coil $\mathrm{B}$ with respect to coil $\mathrm{A}$. Total flux through $\mathrm{B}$,
$$
\mathrm{M}_{21}=\frac{\mathrm{N}_2 \phi_2}{\mathrm{I}_1}
$$
$\therefore$ Magnetic flux $\left(\mathrm{N}_2 \phi_2\right)=10^{-2} \quad \therefore \mathrm{I}_{\mathrm{i}}=2 \mathrm{Amp}$ So, we get
Mutual inductance $=\frac{10^{-2}}{2}=5 \mathrm{mH}$
Now, total flux through $A$ is
$$
\mathrm{N}_1 \phi_1=\mathrm{M}_{12} \mathrm{I}_2=5 \mathrm{mH} \times 1 \mathrm{~A}=5 \mathrm{mWb}\left(\therefore \mathrm{M}_{21}=\mathrm{M}_{12}=5 \mathrm{mH}\right)
$$
$$
\mathrm{N}_2 \phi_2=\mathrm{M}_{21} \mathrm{I}_1
$$
where, $\mathrm{M}_{21}$ is called the mutual inductance of coil A with respect to coil $\mathrm{B}$ and $\mathrm{M}_{21}=\mathrm{M}_{12}$ and $\mathrm{M}_{12}$ is called the mutual inductance of coil $\mathrm{B}$ with respect to coil $\mathrm{A}$. Total flux through $\mathrm{B}$,
$$
\mathrm{M}_{21}=\frac{\mathrm{N}_2 \phi_2}{\mathrm{I}_1}
$$
$\therefore$ Magnetic flux $\left(\mathrm{N}_2 \phi_2\right)=10^{-2} \quad \therefore \mathrm{I}_{\mathrm{i}}=2 \mathrm{Amp}$ So, we get
Mutual inductance $=\frac{10^{-2}}{2}=5 \mathrm{mH}$
Now, total flux through $A$ is
$$
\mathrm{N}_1 \phi_1=\mathrm{M}_{12} \mathrm{I}_2=5 \mathrm{mH} \times 1 \mathrm{~A}=5 \mathrm{mWb}\left(\therefore \mathrm{M}_{21}=\mathrm{M}_{12}=5 \mathrm{mH}\right)
$$
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