# Kirchhoff's Laws and Circuit Analysis

Physics I · Principles of Electromagnetism and Optics · https://tryals.app/learn/physics-i/kirchhoffs-laws-and-circuit-analysis

## Kirchhoff's Laws

Kirchhoff's rules are conservation laws wearing electrical clothing: junctions conserve charge and loops conserve energy.

**Kirchhoff's two laws** form the foundation of circuit analysis:

* **Junction rule (KCL)**: $\sum I_{\text{in}} = \sum I_{\text{out}}$, conserving charge.
* **Loop rule (KVL)**: $\sum \Delta V = 0$ around any closed loop, conserving energy.

| Element | Traversal direction | $\Delta V$ |
| --- | --- | --- |
| Battery ($+$ to $-$) | In direction of EMF | $+\varepsilon$ |
| Battery ($-$ to $+$) | Against EMF | $-\varepsilon$ |
| Resistor | In direction of $I$ | $-IR$ |
| Resistor | Against $I$ | $+IR$ |

**Systematic procedure**: Assign branch currents, write KCL at junctions, write KVL around loops, and solve the linear system.

> **Common pitfall**: Fix a travel direction first and let algebra decide signs. A negative answer just means true current flows opposite your guess.

## RC Circuits and Analysis

An **RC circuit** combines resistors and capacitors, governing how voltage builds over time.

**Charging equation**:

$$V_C(t) = \varepsilon\left(1 - e^{-t/RC}\right)$$

The **time constant** is $\tau = RC$, which is the exact time required to reach $\approx 63\%$ of the final voltage.

**Sanity check**: The number of independent equations always equals the number of unknown currents. If they do not match, you missed a loop or junction.

> **Tip**: Sign errors kill more circuit solutions than concept errors. Always trust your established sign conventions.

## Practice questions

3 of this lesson's 14 practice questions, with answers. The full set is in the app.

### 1. Sort: voltage rise or voltage drop across each element (in direction of conventional current)?

**Answer:**

- Voltage rise: Battery (from − to + terminal), EMF source (traversed from − to +)
- Voltage drop: Resistor (in direction of current), Resistor R with current I: drop = IR, Battery (from + to − terminal), Capacitor being charged (in current direction)

**Why:** Batteries provide voltage rises; resistors and capacitors cause drops.

Page: https://tryals.app/practice/physics-i/kirchhoffs-laws-and-circuit-analysis/sort-voltage-rise-or-voltage-drop-across-each-element-in-direction

### 2. Three identical bulbs: A is in series with the battery; the pair B and C are in parallel with each other after A. If bulb **C burns out** (breaks the branch), what happens to bulb A?

A. A gets dimmer: total resistance rose
B. A gets brighter, the circuit has fewer bulbs to feed
C. A goes out too
D. A is unchanged, it is on its own branch

**Answer:** A. A gets dimmer: total resistance rose

**Why:** Losing C turns the parallel pair ($R/2$) into a single bulb ($R$), raising the loop resistance from $1.5R$ to $2R$. The battery pushes less current, so A dims, while B, now receiving the full branch current, actually brightens.

Page: https://tryals.app/practice/physics-i/kirchhoffs-laws-and-circuit-analysis/three-identical-bulbs-a-is-in-series-with-the-battery-the-pair-b

### 3. Kirchhoff’s two laws are consequences of conservation principles. Why does each of the two rules follow from a conservation law?

A. Both express conservation of energy, at a point and around a loop
B. The junction rule is charge conservation, the loop rule energy
C. Neither: they are empirical rules with no deeper basis
D. The junction rule expresses conservation of energy and the loop rule of charge

**Answer:** B. The junction rule is charge conservation, the loop rule energy

**Why:** Both are conservation laws in circuit clothing. Seeing that is what lets them be applied confidently to circuits that look nothing like the textbook examples.

Page: https://tryals.app/practice/physics-i/kirchhoffs-laws-and-circuit-analysis/kirchhoffs-two-laws-are-consequences-of-conservation-principles-why