- #1
Mark Zhu
- 32
- 3
- 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.
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.
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