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## 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 t

_{0}, this field has value B

_{0}. Over a time interval of 6.8 minutes (hereafter, t

_{1}=6.8 minutes), B changes at a

**constant rate**until a reversed field is reached equal in magnitude to B

_{0}.

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<t

_{1}). 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+B

_{0}

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 t

_{1}, there will be an induced field B

_{i}=B

_{0}. Rearranging our function B(t), we find that B

_{0}=B(t)-Ct. This is equivalent to saying that B

_{i}is equal to the integral of B(t) from 0 to t

_{1}

B

_{i}=∫B(t)=1/2*C*t

_{1}^2

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

C=2B

_{0}/t

_{1}^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 t

_{1}, we find that

Q=CAt

_{1}/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!