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A is primary coil and C is secondary coil

A is connect to a battery while there is no source connected to C

When the current in A varies with rate $\frac{di_0}{dt}$ there is induced emf in B that

$$emf_B=M\frac{di_0}{dt}$$

$emf_B$ cause the current in B and it cause the current to oppose the change of the magnetic flux. That means the current $i_1$ caused by $emf_B$ will create another magnetic field to oppose the magnetic field caused by coil A

The $emf_A=L\frac{di_0}{dt}$ I want to know why the $emf_A$ is not affected by the $emf_B$. enter image description here

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    $\begingroup$ Where's B? I don't see B in your diagram. $\endgroup$ – docscience Jun 15 '15 at 14:00
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I am not sure that your premise is correct.

Imagine the extreme case where the secondary coil is a short-circuited superconductor. In this case, when you try to send a current through the primary coil you will definitely feel a different response than if the secondary was not there.

See for example this tutorial for ways to analyze these kinds of circuits.

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