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Electromagnetism}Faraday Law

Recall Faraday's Law \nabla \times \vec{E} =  -\frac{\partial \vec{B}}{\partial t}. Dot both sides with the area d\vec{A}. Recalling Stokes' Theorem (\int \nabla \times \vec{E} \cdot d\vec{A}=-\frac{\partial \vec{B}}{\partial t}\cdot d\vec{A}), the left side can be converted to the potential, i.e., the emf \mathcal{E}=-\int \vec{E} \cdot d\vec{l}=-\int \nabla \times \vec{E} \cdot d\vec{A}.

Finally, from Ohm's Law V-\mathcal{E}=IR, one can obtain the current. (Note that V=5.0 V is the voltage of the battery. The voltage induced acts to oppose this emf from the battery.)

The problem gives \frac{dB}{dt}=150T/s. The area is just 0.1^2m^2. Thus, the induced emf is,


Thus, V-\mathcal{E}=3.5=IR \Rightarrow I=0.35A, since R=10 \Omega .


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Comments
caffeinated
2008-04-09 15:42:45
By the way Yosun, thanks so much for sharing your knowledge with the rest of us. You've really helped me a lot.NEC
caffeinated
2008-04-09 14:09:33
The induced EMF opposes changes in current and not the constant dc current that is already there. The rate of change in B is negative. This flips the sign on the induced EMF.
caffeinated
2008-04-09 15:15:48
I have just encountered a useful explanation in REA's Physics Problem Solver, Problem 736 on page 721. According to these folks, the direction of the induced current opposes the change in the flux that causes the current in the first place. Since B is decreasing, the induced field is actually in the same direction as B. Pointing the thumb of the right hand in the direction of the induced field, the fingers point in the direction of the induced current, which is clockwise. This has nothing to do with the direction of the current from the DC voltage supply. It just happens to be in the opposite direction. In other words , if the voltage supply is flipped then the currents add. Incidentally, the B field is coming from somewhere else. It is not being caused by the current in the loop, and while there is a small field from the current in the loop, it is negligable compared to the whopping B>>dB=150T/s.
NEC
Richard
2007-09-23 10:39:26
The source alone produces a current, 5[V]/10[\Omega]=.5[A] clockwise.
The change in the magnetic flux \Phi_{B} induces a current, 150[T/s](.01[m^2])/10[\Omega]=.15[A] counterclockwise.
The total current is then, .5[A]-.15[A]=.35[A]
ewhite2
2007-10-24 09:55:55
Remember current flow is opposite the direction of electron flow; it is the flow of 'positive' charges. So the current from the battery flows counterclockwise. Also, since the magnetic field is decreasing in magnitude the induced current will flow clockwise (nature abhors a change in flux).

So the induced current does in fact oppose the current from the battery, however I believe your directions are mixed up.
NEC

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