DC Circuits: EMF, Resistors, Kirchhoff's Rules, and RC Circuits

DC Circuits

Electromotive Force (EMF)

Electromotive Force (EMF) is the energy provided by a battery or other source per unit charge to move charges through a circuit. It is not a force, but a potential difference that drives current.

• Definition: EMF (denoted as $\varepsilon$) is the potential difference maintained by an ideal battery.

• Battery Symbol: The longer line represents the positive (high potential) terminal, and the shorter line the negative (low potential) terminal.

• Work Done: The work done by an ideal battery in moving a charge $q$ is $W = q\varepsilon$.

• Energy Conversion: Batteries convert chemical energy into electrical energy. When chemical reactions cease, the battery can no longer do work.

Example: In a circuit, EMF causes current to flow from the negative to the positive terminal inside the battery, opposite to the direction in the external circuit.

Resistors in Series and Parallel

Resistors are circuit elements that impede the flow of electric current, causing a voltage drop proportional to the current.

Resistors in Series

• Current: The same current flows through all resistors in series.

• Equivalent Resistance: The total resistance is the sum of individual resistances:

$R_{eq} = R_1 + R_2 + \ldots + R_n = \sum_{i=1}^n R_i$

Resistors in Parallel

• Voltage: The voltage drop across each resistor in parallel is the same.

• Equivalent Resistance: The reciprocal of the total resistance is the sum of reciprocals:

$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n} = \sum_{i=1}^n \frac{1}{R_i}$

Example: For $R_1 = 15\,\Omega$, $R_2 = 12\,\Omega$, $R_3 = 24\,\Omega$ with $R_2$ and $R_3$ in parallel, their equivalent resistance is $R_{23} = 8\,\Omega$, and the total resistance in series with $R_1$ is $R_{total} = 23\,\Omega$.

Kirchhoff's Rules

Kirchhoff's rules are fundamental for analyzing complex circuits with multiple loops and junctions.

• Junction Rule: The sum of currents entering a junction equals the sum leaving it (conservation of charge).

• Loop Rule: The sum of voltage changes around any closed loop is zero (conservation of energy).

Application: Assign current directions, write equations for each loop and junction, and solve the resulting system for unknown currents.

Example: Multiloop Circuit Analysis

• Assign polarities to EMF sources.

• Assign currents to each branch.

• Apply Kirchhoff's rules to write equations.

• Solve for unknowns.

Equations:

(1) $I_1 R_1 + I_2 R_2 = \varepsilon_1$ (2) $I_3 R_3 - I_2 R_2 = \varepsilon_2$ (3) $I_1 = I_2 + I_3$

Note: A negative current value indicates the actual direction is opposite to the assumed direction.

Power in DC Circuits

The power dissipated by a resistor is the rate at which it converts electrical energy into heat.

• Formula: $P = I V = I^2 R = \frac{V^2}{R}$

• Battery Power: $P_{battery} = I_{battery} V_{battery}$

Example: For three resistors with currents $I_1$, $I_2$, $I_3$ and voltage drops $V_1$, $V_2$, $V_3$:

$P_1 = I_1 V_1$ $P_2 = I_2 V_2$ $P_3 = I_3 V_3$ $P_{total} = P_1 + P_2 + P_3$

Capacitors in DC Circuits: Charging and Discharging

Capacitors store electrical energy by accumulating charge on their plates. In DC circuits, their behavior is time-dependent.

Charging a Capacitor

• When a switch is closed, current flows and the capacitor charges.

• Kirchhoff's loop rule gives: $\varepsilon - IR - \frac{Q}{C} = 0$

• This leads to a differential equation for $Q(t)$:

$Q(t) = Q_0 (1 - e^{-t/\tau})$ where $Q_0 = \varepsilon C$ and $\tau = RC$ is the time constant.

• The current in the circuit:

$I(t) = \frac{dQ}{dt} = I_0 e^{-t/\tau}$, where $I_0 = \frac{\varepsilon}{R}$

• Voltage across the capacitor:

$V_C(t) = \varepsilon (1 - e^{-t/\tau})$

• Voltage across the resistor:

$V_R(t) = I(t) R$

Discharging a Capacitor

• After the battery is removed, the capacitor discharges through the resistor.

• Kirchhoff's loop rule: $-IR + \frac{Q}{C} = 0$

• Solution for charge:

$Q(t) = Q_0 e^{-t/\tau}$

• Current in the circuit:

$I(t) = I_0 e^{-t/\tau}$

• Voltage across the capacitor:

$V_C(t) = \varepsilon e^{-t/\tau}$

RC Time Constant

• Definition: The time constant $\tau = RC$ characterizes the rate at which the capacitor charges or discharges.

• After a time $t = \tau$, the charge (or voltage) has changed by about 63% of its total change.

Examples and Applications

• Finding Total Resistance: Combine series and parallel resistors using the formulas above.

• Calculating Current: Use Ohm's law $I = V/R$ and Kirchhoff's rules for complex circuits.

• Power Dissipation: Calculate power for each resistor and compare to battery output.

• RC Circuit Analysis: Use exponential equations to find charge, current, and voltage at any time.

Summary Table: Series vs. Parallel Resistors

Configuration	Current	Voltage	Equivalent Resistance
Series	Same through all	Adds across resistors	$R_{eq} = \sum R_i$
Parallel	Splits among branches	Same across all	$\frac{1}{R_{eq}} = \sum \frac{1}{R_i}$

Key Equations

• Ohm's Law: $V = IR$

• Series Resistance: $R_{eq} = \sum_{i=1}^n R_i$

• Parallel Resistance: $\frac{1}{R_{eq}} = \sum_{i=1}^n \frac{1}{R_i}$

• Kirchhoff's Junction Rule: $\sum I_{in} = \sum I_{out}$

• Kirchhoff's Loop Rule: $\sum \Delta V = 0$

• RC Charging: $Q(t) = Q_0 (1 - e^{-t/\tau})$

• RC Discharging: $Q(t) = Q_0 e^{-t/\tau}$

• RC Time Constant: $\tau = RC$

Additional info:

• These notes cover topics from Chapter 26: Direct-Current Circuits and Chapter 24: Capacitance and Dielectrics.

• All equations are in SI units unless otherwise specified.