Calculating Induced Current in a Solenoid-Loop Circuit

[geostudent]

Homework Statement

A cylindrical solenoid 30 cm long with a radius of 8 mm has 400 tightly-wound turns of wire uniformly distributed along its length (see the figure). Around the middle of the solenoid is a two-turn rectangular loop 3 cm by 2 cm made of resistive wire having a resistance of 190 ohms. One microsecond after connecting the loose wire to the battery to form a series circuit with the battery and a 20 resistor, what is the magnitude of the current in the rectangular loop and its direction (clockwise or counter-clockwise in the diagram)? (The battery has an emf of 9 V.)

Homework Equations

B = μ[itex]_{0}[/itex]NI /d

L= μ[itex]_{0}[/itex]N[itex]^{2}[/itex][itex]\pi[/itex]R[itex]^{2}[/itex]/d

I = emf/R * [1-e[itex]^{-(R/L)t}[/itex]]

emf(induced)= d[itex]\Phi[/itex]/dt

The Attempt at a Solution

I just want to check my reasoning here and get advice on how to approach a problem like this.

1. Since the Current is varying with time, I used I = emf/R * [1-e[itex]^{-(R/L)t}[/itex]] to find I at t=1microsecond and got I= .20194 Amperes

2. Used B = μ[itex]_{0}[/itex]NI /d to find B=1.76229E-4 Tesla

3. Induced Emf = -d[itex]\Phi[/itex]/dt where [itex]\Phi[/itex] = ∫B*dA

I took dA to be the cross-sectional Area of the rectangle

Then induced (EMF*Number of turns in rectangle)/R =I

I keep getting relatively close answers but not correct. I don't think I'm thinking of this n the right way. Where am I at fault?

 

[gneill]

The general approach looks alright, although I'm not sure what you're doing with the value of the current in the coil or the value of the B field for that particular time. It'll be the rate of change of the B field that you'll need, no?

Even so, the value you're getting for the current looks a bit odd. What values did you calculate for the inductance and the time constant?