6.
Loop Rule
The second source of equations we will use to solve equations is the Kirchhoff loop rule, which saysThe total change in potential around a closed loop in a circuit is zero.
If we think about potential as height, then this isn't terribly surprising: if you add up all the changes in elevation, positive or negative, as you walk along a loop, of course you will end up at the same elevation you did when you started, and the changes in elevation must add up to zero.
![Changes of potential Changes of potential](Figures/dVchanges.png)
In the circuits we will consider, there are two circuit elements where the electric potential changes.
- Batteries create a potential difference equal to their emf ${\cal E}$, so the positive end of the battery is ${\cal E}$ volts higher than the negative end.
- As current flows through a resistor, its electric potential drops according to Ohm's Law , so the "downstream" end of the resistor is $IR$ volts lower than the "upstream" end.
We can use these two facts to construct an equation, using what I call the tally method:
- Place your finger anywhere on the circuit you wish, and start moving in any direction.
- Everytime you cross a circuit element (battery or resistor), add another term to your equation:
- If you cross a battery from negative to positive ("up the battery"), add $+\cal E$ to your total.
- If you cross a battery from positive to negative ("down the battery"), add $-\cal E$ to your total.
- If you cross a resistor $R$ and your finger is moving in the same direction as the current $I$ ("downstream"), add $-IR$ to your total (using the correct $I$ and $R$ of course).
- If you cross a resistor $R$ and your finger is moving in the opposite direction as the current $I$ ("upstream"), add $+IR$ to your total (using the correct $I$ and $R$ of course).
- Keep tracing the circuit until you return to your starting point. Once you do, set your total equal to zero, and you have your equation.
![(no alternate text) (no alternate text)](Figures/loopall.gif)
The animation shows you the steps to derive three loop rule equations from this circuit:
![(no alternate text) (no alternate text)](Figures/loopcircuit.png)
The loop rules derived are $$\color{red}{8-4I_C-5I_D-7I_A=0}$$ $$\color{blue}{8-9-7I_A=0}$$ $$\color{purple}{+5I_D-6I_E=0}$$