Question

An infinite solenoid has radius b and n turns per unit length. The current varies in time according to $I(t)=I_0cos (\omega t)$ (with positive defined as shown). A ring with radius and resistance R is centred on the solenoid 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 values of t is this force maximum? (c) What is the effect of the force on the ring? That is, does the force cause the ring to translate, spin, ?ip over, stretch/shrink, etc.?

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.