Solenoid Passing through square coil-Finding electric field

[sweetdion]

Solenoid Passing through square coil--Finding electric field

Homework Statement

A very long solenoid with n turns of wire per unit length and radius b carries a current I(t) which decays with time as I(t)=I₀e^-t/T. The direction of current in the solenoid is as shown below. The solenoid passes through a single-turn square coil of wire with side length slightly larger than 2b and resistance R.

a) Determine the electric field both inside and outside the solenoid as the current is decaying. Provide both a magnitude and direction.

b) Determine the current through the square coil as a function of time, I_c(t) as the solenoid current decays.

c) Determine the total energy dissipated in the resistance of the square coil from over the time period t=0 to ∞. Where does the energy come from?

Homework Equations

The induced emf can be obtained from Faraday’s law as
∫ E dr = -d/dt ∫B dA for an open surface S

The Attempt at a Solution

I'm going to start with part a
If we let c=dB/dt, the rate of change of the magnetic field
We choose that surface to be a circle of radius r, and we note by symmetry that the electric field must be tangential to it so for r<=R,
Ein2PiR=cPir²
So Ein=1/2cr tangential to the surface

Eout=2Pir=cPiR²
So Eout=cR2/2r tangential to the surface

I will go onto the next parts next but I'm unsure if this is the right way to find the electric field...

thanks in advance,
sweetdion

 

[quantum13]

although I am a student like yourself, I arrived at the same answers

note that B is obtained by ampere's law and that B =

Spoiler

mu x n x I(t)

and that outside the (ideal) solenoid

Spoiler

B is zero

 

[sweetdion]

so how would we relate the I_c with the square to find part b? This must be Faraday's law...

 

[quantum13]

well I don't think the question is too clear, but I'm assuming the wire somehow jumps onto the square coil then jumps off again onto the solenoid proper..

according to faraday's law, emf = d/dt (flux)

but instead of parsing the electromotive force into its definition as the line integral of electric field, we will now use its other definition as applied to circuitry, namely

Spoiler

I = emf / R

I = dflux/dt / R