Current induced in a solenoid by a constantly changing B

[cbrennan545]

Homework Statement

A circular solenoid with cross section A consists of N turns of wire. The leads of the solenoid are connected to a resistor with resistance R. A magnetic field B is exerted parallel to the normal of the cross section (from left to right, basically). At t₀, this field has value B₀. Over a time interval of 6.8 minutes (hereafter, t₁=6.8 minutes), B changes at a constant rate until a reversed field is reached equal in magnitude to B₀.

How much charge flows through the coil?

(The problem assigns values for each of the variables named above. Notably lacking is a value L describing the length of the solenoid)

Homework Equations

ε=-dΦ/dt
Φ=B⋅A
The magnetic field exerted by a current in a solenoid is B=μnI (n=linear turn density)
V=RI

The Attempt at a Solution

Here is how I thought to solve the problem, but apparently I am wrong:

The big question I need to answer is how much charge flows through the coil (I use the exact wording of the question here, and its a bit vague. I assume it means how much charge flows over the interval 0<t<t₁). I imagine the best way to do this would be to derive some expression for I(t) and integrate.

I start by attempting to solve for B(t), knowing that dB/dt must be constant as constrained in the problem.

dB/dt=C
B=∫(dB/dt)dt=Ct+B₀

Here we invoke Lenz' Law- as a result of a changing external magnetic field, a current will be induced in the solenoid such that the field exerted by the current will directly oppose the change in flux. Hence, at t₁, there will be an induced field B_i=B₀. Rearranging our function B(t), we find that B₀=B(t)-Ct. This is equivalent to saying that B_i is equal to the integral of B(t) from 0 to t₁

B_i=∫B(t)=1/2*C*t₁^2

This allows us to solve for C in term of known variables

C=2B₀/t₁^2

The induced emf, by Faraday's Law

ε=-d/dt(B⋅A)=CA

Invoking Ohm's Law

ε=RI
I=ε/R=CA/R

We now have an expression for I in terms of known variables. Integrating this expression from 0 to t₁, we find that

Q=CAt₁/R, which we can compute.

This is how I have tried to do it, but I can feel deeply in my bones that I did something pretty wrong here but I am emotionally spent from this problem set. One tip off is that I did not use the variable N at any point, which leads me to consider that I likely need to incorporate the expression for the magnetic field of a solenoid somewhere in here. I am hoping someone here can tell me where I went wrong? Thank you so much!

 

[cbrennan545]

I would really appreciate some help with this; the problem set is due at midnight. I'm working on it again right now.

Edit 1: Here is what I have:

I realize that my expression for C might not be correct; I think B_i=ΔB(t)=Ct₁, so

C=B₀/t₁

I did a bit of searching online and found that the actual expression for Faraday's law for a coil is
ε=-N(dΦ/dt)

However, adjusting for these things in my original method is still apparently wrong.

Edit 2: I reread the OP. I call it a "circular" solenoid. I think that might be vague- I mean a normal solenoid with a circular cross section, not a solenoid bent into a circle.