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)
The magnetic field exerted by a current in a solenoid is B=μnI (n=linear turn density)
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.
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 Bi=B0. Rearranging our function B(t), we find that B0=B(t)-Ct. This is equivalent to saying that Bi is equal to the integral of B(t) from 0 to t1
This allows us to solve for C in term of known variables
The induced emf, by Faraday's Law
Invoking Ohm's Law
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!