A long solenoid of initial radius R₀ is put in a region of uniform magnetic field B with the axis of the solenoid aligned along the magnetic field. The solenoid is a part of a closed circuit that has no initial current running through it. If the radius of the solenoid starts increasing at a uniform rate, how do the magnetic field strength B_ in and the associated magnetic energy U_ in inside the solenoid change?

Step-by-step solution

The phrase "closed circuit that has no initial current" implies a perfectly conducting loop where the total magnetic flux remains conserved. Let the external magnetic field be B₀ and the total number of turns be N . The initial magnetic flux through the solenoid is given by: ₀ = N B₀ R₀^2 At a later time, when the radius increases to R , let the net magnetic field inside the solenoid be B_ in . The magnetic flux is: = N B_ in R^2 By the principle of conservation of magnetic flux, = ₀ : N B_ in R^2 = N B₀ R₀^2 B_ in