Question

A square coil of thin wire with 4 turns and side-length 3 cm sits inside an ideal solenoid.  The axis of the solenoid is along +k, but the axis of the coil is tilted at a fixed unknown angle θ from the +z-axis.  The solenoid has n = 5000 turns/meter, and carries current I(t) = 5t³– 4t²+ t + 5 [units of Amperes with t given in seconds] counterclockwise.  The wire of the coil has cross-sectional area A = 50 μm², and the material has a resistivity of ρ = 85 × 10^-8 Ωm

         a.  [5 pts]  What is the resistance of the entire coil?

         b.  [5 pts]  The induced emf in the wire is ε = 3 × 10^-6V at t=0 (ignoring self-induction). What is the angle θ in degrees?

         c.  [5 pts]  What is the magnitude of current in the wire at t = 3s, and does it flow clockwise or counterclockwise around the +k direction?

         d.  [5 pts]  How much power in being delivered to the coil as a function of time?

         e.  [5 pts]  What is the mutual inductance of this configuration?

         f.  [5 pts]  What is the induced emf on the solenoid due to the induced current in the coil, as a function of time?  (Assume that the current in the solenoid is fixed as given, regardless of the induced emf)

Answer 1

a] Length of the coil = 4 x [0.03 + 0.03] = 4.06 m

cross sectional area =

so, the resistance of the entire coil is:

b] From Faraday's law, the emf induced will be due to change in magnetic flux which happens here due to change in current in the coil.

=>

so, at t = 0,

=>

c] At t = 3s,

so, magnitude of induced current is:

.

the current will be clockwise [Lenz's law]

d]

The power delivered to the coil = ..