The formula
- Junction rule, from conservation of charge
ΣI_in = ΣI_out- Loop rule, from conservation of energy
ΣV = 0 around any closed loop- Resistor traversed in the direction of the current
ΔV = -IR- Source traversed from its negative to its positive terminal
ΔV = +ε
What the symbols mean
| Symbol | Meaning | Unit |
|---|---|---|
I | Current in a branch | A |
V | Potential difference | V |
R | Resistance | Ω |
ε | Electromotive force of a source | V |
When it applies
- Circuits with more than one source, or with resistors arranged so that they are neither purely in series nor purely in parallel.
- Assign a direction to every unknown current before you start. A negative answer simply means the real current runs the other way.
- You need as many independent equations as you have unknown currents.
Worked example
Problem. A single loop contains a 12 V source and a 6.0 V source connected in opposition, together with a 4.0 Ω and a 2.0 Ω resistor in series. What current flows?
- Assume the current I runs clockwise, the direction the 12 V source drives it.
- Walk the loop. Through the 12 V source from negative to positive, add 12 V. Through the opposing 6.0 V source from positive to negative, subtract 6.0 V.
- Each resistor is traversed with the current, so each contributes a drop: -I(4.0) and -I(2.0).
- Loop rule: 12 - 6.0 - 4.0I - 2.0I = 0, which gives 6.0 = 6.0I.
- I = 1.0 A, and the positive sign confirms the assumed clockwise direction was right.
Answer. 1.0 A, flowing clockwise.
Common mistakes
- Getting the sign wrong on a resistor. Traversing with the current is a drop; traversing against it is a rise.
- Flipping an assumed current direction partway through the calculation. Keep it and let the sign of the answer correct you.
- Writing loop equations that are not independent, then finding the system cannot be solved.
- Treating a source as a fixed potential difference across the whole branch and ignoring the IR drops that go with it.
Related formulas
- Coulomb's law:
F = k·|q₁q₂| / r² - Faraday's law of induction:
ε = -N·ΔΦ / Δt - Ohm's law:
V = IR