Question

Two inductors of inductance L each are connected in series with opposite magnetic fluxes. The resultant inductance is (Ignore mutual inductance)
A. zero
B. L
C. 2L
D. 3L

Answer 1

Hint: The series-connected inductors will have the net inductance to be equal to the sum of the inductance of the individual inductors. The mutual inductance value should be subtracted from the sum of the inductance of the inductors, if in case given with the value of the mutual inductance.

Formula used:
\[L'={{L}_{1}}+{{L}_{2}}-2M\]

Complete step-by-step answer:
The net inductance will be the sum of the individual inductance of each inductor.
From the data, we the data as follows.
The magnetic flux of both the inductors is opposite in direction to each other.
Let the inductance of the inductors be represented as \[{{L}_{1}},{{L}_{2}}\].
Therefore, the net inductance is given as,
\[L'={{L}_{1}}+{{L}_{2}}-2M\]
As the inductance of both the inductors are given to be the same, that is, L and even, given that, the mutual inductance can be neglected, so, let the value of the mutual inductance be zero.
The direction of the magnetic flux does not have any effect on the net inductance of the two inductors.
So, substitute these values in the above equation.
\[\begin{align}
  & L'=L+L-2(0) \\
 & \Rightarrow L'=2L \\
\end{align}\]
As, the resultant inductance is 2L, thus, the option (C) is correct.

So, the correct answer is “Option (C)”.

Additional Information: If in case, the direction of winding is given.
The winding direction of both the inductors is opposite in direction to each other.
\[\Rightarrow \theta =180{}^\circ \]
According to Faraday’s law, the expression for the inductance is given as follows.
\[e=-L\dfrac{dI}{dt}\]
The emf of the inductors in terms of the inductance of two inductors are as follows:
\[{{e}_{1}}=L\dfrac{dI}{dt}\]
\[{{e}_{2}}=-L\dfrac{dI}{dt}\]
The direction of the emf of the inductors induced will also be in the direction opposite to each other, as the winding direction of both the inductors are opposite in direction to each other.
\[\begin{align}
  & e={{e}_{1}}+{{e}_{2}} \\
 & =L\dfrac{dI}{dt}+\left( -L\dfrac{dI}{dt} \right) \\
 & e=0 \\
\end{align}\]
The negative sign indicates the opposite direction, as this emf opposes the change in the current flowing in the component. Thus, this is often called the back emf.
As the resultant inductance is 2L, thus, the option (C) is correct.

Note: As mentioned above in the additional information, if, in case, instead of the direction of the magnetic flux, the direction of the winding will be given, then, we need to use the emf formula as mentioned above.