# Interaction between a current-driven coil and a second passive coil in the same space

Good evening,I have some doubts about mutual inductance. I hope that somebody will be willing to reply to my questions.

We have a coil L1 traversed by a current I1. If the current is stationary, L1 will generate a magnetic field B according to the Biot-Savart Law. If the current is not stationary, L1 will generate a variable magnetic field according to the Jefimenko's equation. Let's this second case.

Now, I insert a second coil L2 in the same space where B is generated. According to the Faraday's Law, an electromotive force "ε" will be generated in the coil L2. Does the fact that L2 has been inserted in the space of B any sort of effect on the variable current I1?

Then, I connect a load R2 in parallel to the coil L2. A current I2 flow through the circuit L2//R2, how can I determine I2? Does the fact that R2 has been connected to L2 any sort of effect to I1?

Now, I insert a second coil L2 in the same space where B is generated. According to the Faraday's Law, an electromotive force "ε" will be generated in the coil L2. Does the fact that L2 has been inserted in the space of B any sort of effect on the variable current I1?

The effect on $$I_1$$, due to some parasitic capacitance or due to very small eddy currents generated in the bulk of the $$L_2$$ wire, should be negligible.

Then, I connect a load R2 in parallel to the coil L2. A current I2 flow through the circuit L2//R2, how can I determine I2? Does the fact that R2 has been connected to L2 any sort of effect to I1?

Considering that the coils occupy the same space, we can assume that their coupling coefficient is close to $$1$$ and treat the coils as a regular air core transformer, in which case we can use simple relationships between primary and secondary voltages and currents.

The secondary voltage could be found as $$V_2=V_1 \frac {N_2}{N_1}=V_1 \sqrt {\frac {L_2} {L_1}}$$.

When the current starts to flow in $$L_2$$, it produces a flux, which will diminish the flux produced by $$L_1$$, but, assuming the primary coil is connected to the voltage source, it will respond by increasing $$I_1$$, restoring the flux and thus maintaining $$V_2$$ at about the same level it was with an open circuit.

So, the current in the secondary will be $$I_2=\frac {V_2}{R_2}=\frac {V_1}{R}\frac {N_2}{N_1}$$.