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A coil of 40 turns is wrapped around a long solenoid of cross-sectional area 7.5×10−3m2. The solenoid is 0.50 m long and has 500 turns. (a) What is the mutual inductance of this system? (b) The outer coil is replaced by a coil of 40 turns whose radius is three times that of the solenoid. What is the mutual inductance of this configuration?

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To solve this problem it is necessary to apply the concepts related to mutual inductance in a solenoid.

This definition is described in the following equation as,

[tex]M = \frac{\mu_0 N_1 N_2A_1}{l_1}[/tex]

Where,

[tex]\mu =[/tex]permeability of free space

[tex]N_1 =[/tex] Number of turns in solenoid 1

[tex]N_2 =[/tex] Number of turns in solenoid 2

[tex]A_1=[/tex] Cross sectional area of solenoid

l = Length of the solenoid

Part A )

Our values are given as,

[tex]\mu_0 = 4\pi *10^{-7}H/m[/tex]

[tex]N_1 = 500[/tex]

[tex]N_2 = 40[/tex]

[tex]A = 7.5*10^{-4}m^2[/tex]

[tex]l = 0.5m[/tex]

Substituting,

[tex]M = \frac{\mu_0 N_1 N_2A_2}{l_1}[/tex]

[tex]M = \frac{(4\pi *10^{-7})(500)(40)(7.5*10^{-4})}{0.5}[/tex]

[tex]M = 3.77*10^{-4}H[/tex]

PART B) Considering that many of the variables remain unchanged in the second solenoid, such as the increase in the radius or magnetic field, we can conclude that mutual inducantia will appear the same.

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