# Help with Sign of L*dI/dt in an LC circuit

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One concept I always struggle with imparting to my kids in AP Physics C: E&M each year is how to write the sign for the potential drop across an inductor in an RL or LC circuit.  We, of course, always start out with Faraday's Law (and avoid KVL since it is invalid with a changing magnetic flux):

$\oint {\vec E \bullet dl = - {{d{\phi _B}} \over {dt}}}$

But then, when replacing the time rate of change of the magnetic flux through the inductor with L*dI/dt, there's always the question of whether the current is increasing or decreasing, and how best to describe this to students.  If we follow the circuit in a clockwise fashion from the positive side of the battery, the first electric field I see for the voltage source is the negative sign, hence -V on the left-hand-side.  Then, as we follow the circuit the rest of the way around, I end up with the equation:

$- {V_C} = - {Q \over C} = - L{{dI} \over {dt}}$

which is opposite in sign to most of the standard answers I can find.  I have regularly "wiggled around" this problem by telling students to think about which way the current through the inductor is changing to choose your sign, but there has to be a more elegant solution!  This, below, makes sense to me...

Why, then, in problems like AP-C FR EM2 from 2005 as you do the same basic analysis in part B is the sign of LdI/dt reversed?  I struggle with this discrepancy every year.  Any help you can provide would sure be appreciated!

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