What is the Correct Approach for Solving Faraday's Law Problem?

[Mark Zhu]

Homework Statement

Consider the circuit below, which has a capacitor and a resistor and negligible self-inductance. The area enclosed by the circuit is A.
Suppose at t = 0 there is no charge on the capacitor and a magnetic field is switched on which points into the paper. The magnetic field varies with time according to
|B→| = B[SUB]0[/SUB] sin(ωt)
where B[SUB]0[/SUB] and ω are constants. Find the charge on the top plate as a function of time.

Relevant Equations

Q = CV
V = IR

This seems like just another Faraday's Law problem, but I'm getting the wrong answer according to the book. I think I'm only calculating the answer for the interval ωt = 0 and ωt = pi/2, when the |B→| is increasing. Basically you just calculate the magnetic flux through the area of the loop, which is -B₀Asin(ωt). It's negative because I have chosen to go CCW around the loop, making dA→ point out of the page while the magnetic field points inwards. Taking the derivative of this WRT time is just -B₀Aωcos(ωt). I set this equal to the negative closed path integral of E→ ⋅ dr→, which is -Q/C-iR since I'm going ccw around the path in the same direction as the current. After doing some math and noting Q(0) = 0, I get that Q(t) = CB₀Aωcos(ωt)(1-e^-t/(RC)).
However, the book has a super complex and weird answer and with 3 terms: 1 with cos, 1 with sin, and 1 with e exponential.
Do I only have half the answer since I only considered the interval between ωt = 0 and ωt = pi/2, where |B→| is increasing? Thanks a lot.

 

[doggydan42]

After doing some math and noting Q(0) = 0, I get that Q(t) = CB₀Aωcos(ωt)(1-e^-t/(RC)).

How did you solve the differential equation for Q(t)?

 

[ehild]

. After doing some math and noting Q(0) = 0, I get that Q(t) = CB₀Aωcos(ωt)(1-e^-t/(RC)).
However, the book has a super complex and weird answer and with 3 terms: 1 with cos, 1 with sin, and 1 with e exponential.

Check if your function is really solution of the problem.

 

[Mark Zhu]

How did you solve the differential equation for Q(t)?

 

[doggydan42]

On the line where you introduced the natural logarithm, there is a cosine term of t, so you cannot simply say that it is an integral like $\frac{dQ}{Q+C}$, for some constant C. Separation of variables does not work because you have that cos(ωt).

You need to solve the homogeneous differential equation first to get the complementary solution, then you can make an ansatz for the full differential equation to get the particular solution.

 

[Mark Zhu]

On the line where you introduced the natural logarithm, there is a cosine term of t, so you cannot simply say that it is an integral like $\frac{dQ}{Q+C}$, for some constant C. Separation of variables does not work because you have that cos(ωt).

You need to solve the homogeneous differential equation first to get the complementary solution, then you can make an ansatz for the full differential equation to get the particular solution.

Thank you so much