One solenoid is centered inside another. The outer one has a length of 54.0 cm and contains 6750 coils, while the coaxial inner solenoid is 4.00 cm long and 0.170 cm in diameter and contains 21.0 coils. The current in the outer solenoid is changing at 35.0 A/s .What is the mutual inductance of the solenoids?Find the emf induced in the inner solenoid.

 M₁₂ = 1.01 10⁻⁴ H ,   Fem = 3.54 10⁻³ V

Explanation:

The mutual inductance between two systems is

        M₁₂ = N₂ Ф₁₂ / I₁

where N₂ is the number of turns of the inner solenoid N₂ = 21.0, i₁ the current that flows through the outer solenoid I₁ = 35.0 A / s and fi is the flux of the field of coil1 that passes through coil 2

the magnetic field of the coil1 is

   B = μ₀ n I₁ = μ₀ N₁/l   I₁

the flow is

             Φ = B A₂

the area of the second coil is

             A₂ = π d₂ / 4

             Φ = μ₀ N₁ I₁ / L  π d² / 4

we substitute in the first expression

            M₁₂ = N₂ μ₀ N₁ / L    π d² / 4

            M₁₂ = μ₀ N₁ N₂ π d² / 4L

           d = 0.170 cm = 0.00170 m

            L = 4.00 cm = 0.00400 m

let's calculate

            M₁₂ = 4π 10⁻⁷ 6750  21 π 0.0017²/ (4 0.004)

             M₁₂ = π² 0.40966 10⁻⁷ / 0.004

             M₁₂ = 1.01 10⁻⁴ H

The electromotive force is

              Fem = - M dI₁ / dt

              Fem = - 1.01 10⁻⁴ 35.0

              Fem = 3.54 10⁻³ V

representative si the final answer! : )