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- Homework Statement:
- The circuit in the figure below is located in a magnetic field whose magnitude varies with time according to the expression ##B=1.00\times10^{-3}t##, where ##B## is in teslas and ##t## is in seconds. Assume the resistance per length of the wire is ##0.094## Ω/m. Find the magnitude of the current in section ##PQ## of length ##a = 55.0## cm, and its direction.

- Relevant Equations:
- $$\mathcal{E}=-\phi_B'(t)$$

I have nearly finished my homework, but am blocking at one of the simplest questions...

I'm going to let ##\rho=0.094\,\Omega/m##

Since the magnetic flux is increasing, then, by Lenz's law, the current in both loops is counter-clockwise.

If we only look at the left loop, then ##\phi_L=2a^2B\Leftrightarrow\mathcal{E}=-2a^2B'(t)##. The resistance is ##R=6a\rho##, so ##I_L=-\frac{aB'(t)}{3\rho}\approx−0.00195\,A## (L for left).

If we only look at the right loop, then ##\phi_R=a^2B\Leftrightarrow\mathcal{E}=-a^2B'(t)##. The resistance is ##R=4a\rho##, so ##I_R=-\frac{aB'(t)}{4\rho}\approx−0.00146\,A##.

Since ##I_L## comes to ##PQ## from ##Q##, and is more negative than ##I_R##, then the direction of the total current is from ##Q## to ##P##, and ##|I|=I_R-I_L##.

The resultant magnitude is wrong, but the direction of the current is correct. Can anyone guide me a bit? Thank you very much!

I'm going to let ##\rho=0.094\,\Omega/m##

Since the magnetic flux is increasing, then, by Lenz's law, the current in both loops is counter-clockwise.

If we only look at the left loop, then ##\phi_L=2a^2B\Leftrightarrow\mathcal{E}=-2a^2B'(t)##. The resistance is ##R=6a\rho##, so ##I_L=-\frac{aB'(t)}{3\rho}\approx−0.00195\,A## (L for left).

If we only look at the right loop, then ##\phi_R=a^2B\Leftrightarrow\mathcal{E}=-a^2B'(t)##. The resistance is ##R=4a\rho##, so ##I_R=-\frac{aB'(t)}{4\rho}\approx−0.00146\,A##.

Since ##I_L## comes to ##PQ## from ##Q##, and is more negative than ##I_R##, then the direction of the total current is from ##Q## to ##P##, and ##|I|=I_R-I_L##.

The resultant magnitude is wrong, but the direction of the current is correct. Can anyone guide me a bit? Thank you very much!

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