Sign of the terms in a circuit

[Silviu]

Hello! I am a bit confused about the sign of the $L\frac{dI}{dt}$ term in the circuits (DC circuits). In my book it is defined with a minus, on wikipedia it is defined with a plus and I am not sure which one should I use. I can pick any sign I want and the result will come out right regardless? Won't I risk to get a wrong ODE if I have a different sign for that term? Thank you!

 

[Charles Link]

Yes. The correct differential equation is $\mathcal{E}=L \frac{dI}{dt}+IR +\frac{Q}{C}$, where $I=\frac{dQ}{dt}$. $\\$ The inductor can be considered to be a voltage source/EMF with voltage $\mathcal{E}_{inductor}=-L \frac{dI}{dt}$, which can be put on the left side of the differential equation. Alternatively, it can be considered an element with a voltage drop of $L \frac{dI}{dt}$ just like a resistor has a voltage drop of $IR$. The voltage drop is positive when the current is increasing.

 

[Silviu]

Yes. The correct differential equation is $\mathcal{E}=L \frac{dI}{dt}+IR +\frac{Q}{C}$, where $I=\frac{dQ}{dt}$.

Thank you! So here $\epsilon$ is the sum of any voltage source in the circuit? Something like $\epsilon = \sum_i V_i$

 

[Charles Link]

Yes. And I get the symbol with \mathcal{E} surrounded by the Latex parameters.$\\$ And please see the last couple of sentences I added to post 2.

 

[vanhees71]

Well the signs you understand best when going back to the first principles the circuit equations derive from. For induction it's of course Faraday's law, and there (as with all things having to do with curls and rotation) the right-hand rule is your friend. Here are some notes from a lecture, I've given to Texan engineering students some time ago. Maybe they are of some use (although in my very bad handwriting ;-():

https://th.physik.uni-frankfurt.de/~hees/physics208/phys208-notes-III.pdf

The part on induction you starts on page 106. The course website with some more material is here:

https://th.physik.uni-frankfurt.de/~hees/physics208.html