Now find the electromotive force e2(t) induced across the entirety of solenoid 2 by the change in current in solenoid 1. remember that both solenoids have length l.

Explanation:

Consider two solenoids that are wound on a common cylinder as shown in fig. 1. Let the cylinder have radius 'ρ' and length 'L' .

No. of turns of solenoid 1 = n₁

No. of turns of solenoid 1 = n₂

Assume that length of  solenoid is much longer than its radius, so that its field can be determined from Ampère's law throughout its entire length:

We will consider the field that arises from solenoid 1, having n₁  turns per unit length. The magnetic field due to solenoid 1 passes through solenoid 2, which has n₂ turns per unit length.

Any change in magnetic flux from the field generated by solenoid 1 induces an EMF in solenoid 2 through Faraday's law of induction:

Consider B₁(t) magnetic feild generated in solenoid 1 due to current I₁(t)

Using:

                                  --- (2)

Flux generated due to magnetic field B₁

                      ---(3)

area of solenoid =

substituting (2) in (3)

                        ----(4)

We have to find electromotive force E₂(t) induced across the entirety of solenoid 2 by the change in current in solenoid 1, i.e.

                        ---- (5)

substituting (4) in (5)

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