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If the induced emf in a circuit is negative, and current from this emf is the emf over the resistance, what happens to the negative sign in the induced emf when solving for the current? Surely there's no such thing as "negative" current?

For example, if we have a constant magnetic field that is pointing down, and the area of a loop is increasing (so that the flux is positive), then the emf will be negative. But if the magnetic field is pointing down, then since the emf opposes a change, it will point up and the current will go counterclockwise. So is this what the negative current is accounting for?

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    $\begingroup$ If the plus and minus signs are confusing, you can just use Lenz's Law. en.wikipedia.org/wiki/Lenz's_law $\endgroup$
    – Mark Eichenlaub
    Commented Dec 10, 2010 at 8:16
  • $\begingroup$ @Mark yes but if I use it does it mean I should ignore the signs? $\endgroup$
    – wrongusername
    Commented Dec 10, 2010 at 8:18
  • $\begingroup$ Why the flux is positive? $\endgroup$
    – kennytm
    Commented Dec 10, 2010 at 8:22
  • $\begingroup$ @Kenny because area is increasing in a constant B field $\endgroup$
    – wrongusername
    Commented Dec 10, 2010 at 8:24
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    $\begingroup$ @wrongusername Personally, I would use Lenz's law to work out the direction of the current, then use that to double-check the work I did with the signs. You don't have to ignore the signs, since everything will technically work out fine, but you can check your answer that way. If you do rely on Lenz's law and ignore the signs, that should actually work out fine, too. It might be considered a little sloppy by a TA, though. $\endgroup$
    – Mark Eichenlaub
    Commented Dec 10, 2010 at 8:30

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Negative current just means it flows opposite to the chosen direction.

Take your example, you have a loop with increasing area and downward pointing B-field

x   x  x  x  x  x  x  x  x
  +-------------|-----------  ^
x | x  x  x  x  |  x  x  x    |     y
  |             | --->        h     ^
x | x  x  x  x  |  x  x  x    |     |
  +-------------|-----------  v     o---> x
x   x  x  x  x  x  x  x  x         z

  <--- w(t) ---->

Therefore, the flux is $\Phi = hw(t)\mathbf B\cdot \hat{\mathbf z} = -Bhw(t)$ which is negative. Since the area is increasing, $dw/dt > 0$, so $d\Phi/dt = -Bh\,dw/dt < 0$ is also negative. The emf $\mathcal E = -d\Phi/dt > 0$ is thus positive, and the induced current flows counterclockwise as expected.

The sign of flux doesn't just depend on the area, but also the direction of the B-field relative to the surface normal.

As suggested by Mark in the comments, it would be easier to determine the sign using Lenz's law.

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  • $\begingroup$ I updated my question with a reply :) $\endgroup$
    – wrongusername
    Commented Dec 10, 2010 at 8:14
  • $\begingroup$ There is an instance where opposing voltages prevent any current change, when change of current is attempted by any means (Lenz's law) energy stored in the magnetic field has to be converted to potential energy (voltage) whenever you interrupt or alter the current. You can get a shock from a 1.5 volt cell connected to a reasonably large iron cored coil if you connect for a second or two and then disconnect the cell, the high voltage will appear momentarily across the coil. The voltage can reach 200 volts or more depending on the coil. $\endgroup$
    – barry
    Commented May 19, 2021 at 13:12

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