Kirchhoff's Loop Rule
Introduced| English |
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| Kirchhoff's loop rule/ˈkɜːkhɒfs luːp ruːl/ |
Walk a full loop and you return to the same energy
- Follow one charge all the way around a circuit loop.
- It gains energy at the battery and loses it at each resistor.
- Back at the start, its energy is exactly what it began with.
- That simple bookkeeping is the loop rule — a tool for any circuit.
The loop rule: voltages sum to zero
- Around any closed loop, the voltage changes add to zero: $\sum V = 0$.
- Every rise (battery) is matched by the drops (resistors).
- For a single loop: $\varepsilon - IR_1 - IR_2 = 0$.
- It is really energy conservation, counted per unit charge.
Kirchhoff's loop rule states that around any closed loop:
$\sum V = 0$ around a loop — energy conservation per charge.
A $12\ \text{V}$ battery drives current through a $2\ \Omega$ and a $2\ \Omega$ resistor in series. Find $I$ (in A).
$12 - I(2) - I(2) = 0 \Rightarrow 12 = 4I \Rightarrow I = 3\ \text{A}$.
Getting the signs right
- Going through a battery from $-$ to $+$ is a rise ($+\varepsilon$).
- Crossing a resistor with the current is a drop ($-IR$).
- Cross it against the current and it counts as a rise ($+IR$).
- Pick a direction, walk the loop, and keep the signs consistent.
Crossing a resistor in the direction of the current is a voltage drop ($-IR$).
With the current, a resistor drops the potential by $IR$.
Select all true statements about the loop rule.
Loop rule: ΣV = 0, energy conservation, consistent signs. Currents-sum-to-zero is the junction rule.
Why it must be true
- A charge returning to its start is at the same potential.
- So the total potential change around the loop is zero.
- Energy given by the source equals energy taken by the components.
- No loop can create or destroy energy for the charge.
Kirchhoff's loop rule
Around any closed loop the voltage rises equal the voltage drops. Sort each crossing.
The loop rule is a statement of the conservation of:
A charge returns to the same potential — energy is conserved.
Using it to solve circuits
- Write one loop equation for each independent loop.
- Combine with the junction rule to get enough equations.
- Solve them together for the unknown currents.
- Complex networks fall to a handful of linear equations.
The loop rule is combined with the ____ rule to solve multi-loop circuits.
The junction rule provides the extra current equations.
A $9\ \text{V}$ battery drives one current through a $2\ \Omega$ and a $1\ \Omega$ resistor in series.
- Loop rule: $9 - I(2) - I(1) = 0$.
- So $9 = 3I$, giving $I = 3\ \text{A}$.
Keep your sign convention consistent. Decide a loop direction first, then a resistor crossed with the current is $-IR$ and against it is $+IR$. Flipping signs midway is the usual source of wrong answers.
Kirchhoff's loop rule 基尔霍夫回路定则 says the voltage changes around any closed loop sum to zero: $\sum V = 0$. It is energy conservation per charge — a battery rise balances the resistor drops. Track signs carefully and pair it with the junction rule.