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- Jan 26, 2012

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*Physics, 6th Ed.*, which goes like this:

A closed conducting circle of wire is lying the plane of the page (or screen), and there is a magnetic $\mathbf{B}$ field pointing to the left, in the plane of the page (or screen). Hence, there is initially zero magnetic flux through the area of the circle. Then we rotate the loop about a vertical axis (in the plane of the page) by pulling the left side towards us out of the page and pushing the right side into the page. In which direction is the induced current: clockwise or counter-clockwise? See the following picture for an illustration of the physical situation.

There are several ways to think about this. Probably the most intuitive method is the conservation of energy approach: the induced current must produce a magnetic field which does not, in turn, contribute to the magnetic force already on the loop. If it did, we would have a positive feedback loop situation, and we could get a perpetual motion machine. Therefore, the current must produce a magnetic field in the area of the loop that opposes the existing magnetic field. Hence, the current must be counterclockwise.

However, the standard method of thinking about this problem is Lenz's Law, which states that a current induced by a changing magnetic flux must oppose that change. Now the magnetic flux we define to be $\Phi_{B}=BA\cos(\theta)$, where $\theta$ is defined to be the angle between the normal vector (which one?!?) and the magnetic field. If we say that the normal vector is initially pointing

*into*the page, then the angle between the normal and the magnetic field is decreasing, the $\cos(\theta)$ is increasing (going from $0$ to $1$), and hence the current must generate a magnetic field that opposes the existing magnetic field. Hence, it must be counterclockwise. This agrees with the previous result.

But now, suppose we define the normal vector to be out of the page. The angle $\theta$ would be getting larger (more obtuse), and hence the $\cos(\theta)$ would be getting more negative. This was the situation that was confusing me in class. If the flux is getting more negative, then the induced current must be trying to make the flux less negative. Hence, the induced current must be trying to make the flux more positive. Which current direction would make that happen? Here you use the right-hand rule to fix the problem: if the current were to go counter-clockwise, your right hand would be curling around in that direction, and your thumb would point out of the page

*in the same direction as the normal force*. That's positive. But, if you had a clockwise current, your thumb would point into the page, which is negative according to the way you defined the normal force.

So, counter-clockwise it is, then. Too bad that neither Giancoli nor Holt Physics explains this. I'll have to double-check Young and Freedman as well as Serway for calc-based books when I get back to school.