**Derivation of Mutual Inductance of Two Long Solenoids**

To derive the mutual inductance of two long solenoids, we can consider the magnetic field produced by one solenoid and calculate the magnetic flux through the other solenoid. By using Faraday's law of electromagnetic induction, we can then relate the induced emf to the rate of change of magnetic flux. This will allow us to determine the mutual inductance between the two solenoids.

**Magnetic Field Produced by One Solenoid**

The magnetic field produced by a long solenoid is given by the formula:

B = μ₀nI

Where B is the magnetic field, μ₀ is the permeability of free space, n is the number of turns per unit length, and I is the current flowing through the solenoid.

**Magnetic Flux through the Second Solenoid**

Consider two long solenoids, labeled as Solenoid 1 and Solenoid 2, placed parallel to each other. We want to calculate the magnetic flux through Solenoid 2 due to the magnetic field produced by Solenoid 1.

The magnetic flux (Φ₂) through Solenoid 2 can be calculated by integrating the magnetic field (B₁) produced by Solenoid 1 over the cross-sectional area (A₂) of Solenoid 2:

Φ₂ = ∫∫ B₁ · dA₂

**Induced EMF and Mutual Inductance**

According to Faraday's law of electromagnetic induction, the induced electromotive force (emf) in a circuit is equal to the negative rate of change of magnetic flux through the circuit.

Therefore, the induced emf (ε₂) in Solenoid 2 due to the changing magnetic flux through it can be given by:

ε₂ = -dΦ₂/dt

To relate the induced emf to the rate of change of magnetic flux, we introduce the concept of mutual inductance (M) between the two solenoids. Mutual inductance is a measure of how much of the magnetic field produced by one solenoid threads through the other.

The mutual inductance (M) between Solenoid 1 and Solenoid 2 is given by:

M = Φ₂/I₁

Where I₁ is the current flowing through Solenoid 1.

Therefore, the induced emf in Solenoid 2 due to the changing current in Solenoid 1 can be expressed as:

ε₂ = -M(dI₁/dt)

Hence, by calculating the magnetic flux through Solenoid 2 and using Faraday's law, we can determine the mutual inductance between the two long solenoids.