Tally Method

Consider the circuit shown, with a 3V battery and three resistors. A 0.5A current flows through the circuit, and I want to know the potential difference between the points i (for initial) and f (final) shown. How do I go about it?

This is the potential \(V_f\), so the potential difference is \(\Dl V=V_f-V_i=(V_i+1.5\u{V})-V_i=\boxed{1.5\u{V}}\)

The tally method is a way of standardizing this calculation. Imagine a small bug (represented here by a purple dot) which travels along the wires of a circuit, keeping track of changes in the potential as it goes. To find the potential difference between two points on a circuit:

  1. Start the bug at the initial position, and begin moving towards the final position along any path.
  2. If the bug crosses a battery from negative to positive, add the battery's emf \(\cal E\) to the tally. (We'll call this "moving up" the battery.) If the bug "moves down" the battery, subtract the emf from the tally.
  3. If the bug moves through a resistor \(R\) in the same direction as the current \(I\) (we can call this "downstream" or "with the current"), subtract \(IR\) from the tally. If it moves upstream, add \(IR\) to the tally.
  4. When the bug reaches the end position, the tally is the potential difference \(\Dl V\) between the points. The final potential \(V_f\) is the tally added to the initial potential: \(V_f=V_i+\Delta V\).
Interactive 11.2.2
In the circuit shown, the point in the upper-left corner has a potential has 3V. What is the potential at the lower-right corner?

(A) We start at \(V=3\u{V}\). If we go clockwise around the loop, we go down the 1Ω resistor, which is a drop of \((0.25\u{A})(1\u{\Omega})=0.25\u{V}\). We then go down the 3Ω resistor for an additional drop of 0.75V, for a total drop of 1V. Thus the potential at the bottom-right corner is 3V-1V=2V. Show this on the circuit.

(B) We can also go counterclockwise around the loop. We go down the battery, so the potential drops by 2V. We then go up the 4Ω resistor, which adds \((0.25\u{A})(4\u{\Omega})=1\u{V}\) to the tally. The potential at the bottom corner is $$3\u{V}-2\u{V}+1\u{V}=\boxed{2\u{V}}$$ which is the same as before. Show this on the circuit.


Loop Rule

If the bug travels through the circuit and returns to its starting point, then its final potential is clearly the same as its initial potential, and so the tally must be zero. You can confirm this for yourself in the previous example: if the bug starts in the lower-left corner and goes clockwise, then the tally is $$+2-(0.25)(1)-(0.25)(3)-(0.25)(4)=0$$ If we don't know something about a circuit---the current through a wire, or the resistance in a resistor---we can use this fact to solve for the unknown.
Find the current \(I\) in this circuit, using the tally method around a closed loop.
Start your bug in the upper-right corner, and go clockwise. The bug goes down both resistors and then up the battery before returning to its starting point. The tally is $$-(2\u{\Omega})I-(3\u{\Omega})I+4\u{V}$$ Because the bug completed a loop, this tally must be equal to zero: $$\begin{eqnarray} -2I-3I+4&=&0\\ \implies 4&=&5I\\ \implies I&=&\boxed{0.8\u{A}}\\ \end{eqnarray}$$