Question
Obtain expression for induced emf in any one coil due to electric currents flowing at same time in two coils placed near each other.
✓
Answer
→When electric currents are passed simultaneously through two coils placed near each other, magnetic flux linked with any one coil is sum of individual magnetic flux linked independently with both coils.
→Suppose, electric currents passing simultaneously through two nearby coils having turns $N _1$ and $N _2$ is $I _1$ and $I _2$ respectively.
→Total magnetic flux linked with coil having $N _1$ turns is
$\therefore N _1 \phi_1= M _{11} I _1+ M _{12} I _2$
Where $M _{11}$ inductance due to first coil
$\begin{array}{ll}
\therefore \quad & M _{11}= L _1 \quad \text { and } \\
& M _{12}=\text { mutual inductance of first coil } \\
& \text { with respect to second coil } \\
\therefore \quad N _1 \phi_1= & L _1 I _1+ M _{12} I _2
\end{array}$
According to Faraday's law, induced emf in first coil,
$\begin{aligned}
\varepsilon_1 & =- N _1 \frac{d \phi_1}{d t} \\
\therefore \varepsilon_1 & =- L _1 \frac{d I _1}{d t}- M _{12} \frac{d I _2}{d t}
\end{aligned}$
Equation (3) is required expression.
→Suppose, electric currents passing simultaneously through two nearby coils having turns $N _1$ and $N _2$ is $I _1$ and $I _2$ respectively.
→Total magnetic flux linked with coil having $N _1$ turns is
$\therefore N _1 \phi_1= M _{11} I _1+ M _{12} I _2$
Where $M _{11}$ inductance due to first coil
$\begin{array}{ll}
\therefore \quad & M _{11}= L _1 \quad \text { and } \\
& M _{12}=\text { mutual inductance of first coil } \\
& \text { with respect to second coil } \\
\therefore \quad N _1 \phi_1= & L _1 I _1+ M _{12} I _2
\end{array}$
According to Faraday's law, induced emf in first coil,
$\begin{aligned}
\varepsilon_1 & =- N _1 \frac{d \phi_1}{d t} \\
\therefore \varepsilon_1 & =- L _1 \frac{d I _1}{d t}- M _{12} \frac{d I _2}{d t}
\end{aligned}$
Equation (3) is required expression.
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