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

DC Circuits

Electromotive Force (EMF)

Electromotive Force (EMF) is a fundamental concept in electric circuits, representing the energy provided by a source (such as a battery) to move charge through a circuit.

• Definition: EMF is the potential difference generated by a source when no current is flowing.

• Circuit Symbol: The battery symbol shows the high potential at the longer line and low potential at the shorter line.

• Direction of Current: Inside the battery, current flows from low to high potential; in the external circuit, it flows from high to low potential.

• 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: A battery with EMF $\varepsilon$ does $W = q\varepsilon$ work to move charge $q$ through the circuit.

Resistors in Series and Parallel

Resistors are circuit elements that impede the flow of electric current, and their arrangement affects the total resistance and current distribution in a circuit.

Resistors in Series

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

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

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

• Voltage Drop: The total voltage drop is the sum of the drops across each resistor.

Resistors in Parallel

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

• Current: The total current splits among the parallel branches.

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

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

• Conservation of Charge: At a junction, $I = I_1 + I_2 + \ldots$

Example: For three resistors $R_1 = 15\,\Omega$, $R_2 = 12\,\Omega$, $R_3 = 24\,\Omega$ with $R_2$ and $R_3$ in parallel, their equivalent resistance is:

$R_{23} = \frac{R_2 R_3}{R_2 + R_3} = \frac{12 \times 24}{12 + 24} = 8\,\Omega$

Total resistance: $R_{total} = R_1 + R_{23} = 15 + 8 = 23\,\Omega$

Kirchhoff's Rules

Kirchhoff's rules are essential 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 and EMF polarities, then write equations for each loop and junction to solve for unknowns.

Example: Multiloop Circuit

Given a circuit with three resistors and two batteries, the following equations can be written:

• Loop 1: $\varepsilon_1 - I_1 R_1 - I_2 R_2 = 0$

• Loop 2: $\varepsilon_2 + I_2 R_2 - I_3 R_3 = 0$

• Junction: $I_1 = I_2 + I_3$

Solving these equations yields the current in each branch. Negative current values indicate 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: $P_{battery} = I_{battery} \Delta V_{battery}$

Example: For $I_1 = 0.435\,A$ and $V_1 = 6.53\,V$, $P_1 = I_1 V_1 = 2.84\,W$

Capacitors in DC Circuits: Charging and Discharging

Capacitors store electrical energy and exhibit time-dependent behavior when charging or discharging through a resistor.

Charging a Capacitor

• Differential Equation: $\varepsilon - IR - \frac{Q}{C} = 0$

• Solution: $Q(t) = Q_0 (1 - e^{-t/\tau})$ where $Q_0 = \varepsilon C$ and $\tau = RC$

• Current: $I(t) = \frac{dQ}{dt} = I_0 e^{-t/\tau}$

• Voltage across Capacitor: $V_C(t) = \varepsilon (1 - e^{-t/\tau})$

• Voltage across Resistor: $V_R(t) = I(t) R$

Discharging a Capacitor

• Differential Equation: $-IR + \frac{Q}{C} = 0$

• Solution: $Q(t) = Q_0 e^{-t/\tau}$

• Current: $I(t) = I_0 e^{-t/\tau}$

• Voltage across Capacitor: $V_C(t) = \varepsilon e^{-t/\tau}$

Time Constant: $\tau = RC$ determines how quickly the capacitor charges or discharges.

Example: RC Circuit Analysis

• Given $C = 1.00\,\mu F$, $R = 2.00 \times 10^6\,\Omega$, $\varepsilon = 10.0\,V$, at $t = 10.8\,s$:

• Time constant: $\tau = RC = 2.00\,s$

• Charge: $Q(t) = C\varepsilon (1 - e^{-t/\tau}) = 9.95 \times 10^{-6}\,C$

• Current: $I(t) = I_0 e^{-t/\tau} = 2.26 \times 10^{-8}\,A$

• Power stored: $P = \frac{dU}{dt} = \frac{Q}{C} \frac{dQ}{dt}$

• Power delivered by battery: $P = I\varepsilon$

Summary Table: Series vs. Parallel Resistors

Property	Series	Parallel
Current	Same through all resistors	Splits among branches
Voltage	Divides among resistors	Same across all resistors
Equivalent Resistance	$R_{eq} = \sum_{i=1}^n R_i$	$\frac{1}{R_{eq}} = \sum_{i=1}^n \frac{1}{R_i}$

Key Equations

• Series: $R_{eq} = R_1 + R_2 + \ldots + R_n$

• Parallel: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots + \frac{1}{R_n}$

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

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

• RC Time Constant: $\tau = RC$

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

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

Steps for Solving Multiloop Circuit Problems

1. Assign polarity (+/–) to all EMF sources.

2. Assign currents to each branch of the circuit.

3. Apply Kirchhoff's rules to write equations for loops and junctions.

4. Solve the resulting system of equations for the unknowns.

Additional info:

• Negative current values indicate the actual direction is opposite to the assumed direction.

• All energy stored in a capacitor is eventually dissipated by the resistor during discharge.