Question

An infinite solenoid with a radius b has n turns per unit length. The current varies in terms according to I(t) =Iocoswt (with wpositive defined as shown in the below, i. e counterclockwise) A ring with radius r < b and resistance R is centered on the solenoid's axis, with its plane perpendicular to the axis. a) What is the induced current in the ring? 

b) A given little piece of the ring will feel a magnetic force. For what value of t is this force maximum. 

C) What is the effect of the force on the ring? In other words, does the force cause the ring to translate, spin, flip over, stretch or shrink?

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Answer 1

a) The magnetic field inside the solenoid is, (B) = mu_0 nI(t) The induced emf is given by, epsilon = d phi/dt = -d/dt(BA) epsilon = -d/dt(mu_0 nI(t) times pi r^2) epsilon = -mu_0 n pi r^2 times dI(t)/dt epsilon = -mu_0 n pi r^2 times d/dt(I_0 cos(Omega t)) epsilon = -mu_0 n pi r^2 I_0 Omega(-sin(Omega t)) epsilon = mu_0 n pi r^2 I_0 Omega sin(Omega t) Therefore, the required current is, I = epsilon/R I = mu_0 n pi r^2 I_0 Omega/R sin(Omega t) b) For the magnetic force, it will have the contribution from both the magnetic field and the current. Therefore F = constant times cos(omegat) times sin(Omega t) F = constant times 1/2 times cos(Omega t) times sin(Omega t) F = C times sin(2omega t) Therefore For maximum force, the condition is sin(2omega t) = 1 2omega t = pi/2, 3pi/2---- Therefore t = pi/4omega---- Therefore, the required value of t is t = pi/4omega, 3pi/4omega--- c) The force will produce a torque and the ring will spin.