Direct-Current Circuits: Resistance, EMF, and Circuit Analysis

Direct-Current Circuits

Introduction

This section covers the fundamental principles of direct-current (DC) circuits, focusing on current, resistance, resistivity, electromotive force (emf), power, and the analysis of circuits containing resistors and capacitors. The material is essential for understanding how electrical circuits operate and how to analyze their behavior using physical laws.

Current and Current Density

Definition and Concepts

• Electric Current (I): The rate at which electric charge flows through a conductor. Measured in amperes (A).

• Current Density (J): The current per unit cross-sectional area of a conductor. It is a vector quantity, given by $\vec{J} = \frac{I}{A}$, where $A$ is the area.

• Direction: By convention, current direction is the direction positive charges would move.

Example: In a wire carrying 2 A of current with a cross-sectional area of $1 \text{ mm}^2$, the current density is $J = 2 \times 10^6 \text{ A/m}^2$.

Resistance and Resistivity

Definitions

• Resistance (R): The opposition to the flow of electric current, measured in ohms ($\Omega$). For a resistor, $V = IR$.

• Resistivity ($\rho$): A material property that quantifies how strongly a material opposes current. $R = \rho \frac{l}{A}$, where $l$ is length and $A$ is cross-sectional area.

Example: A copper wire of length 2 m and area $1 \text{ mm}^2$ has resistance $R = \rho \frac{l}{A}$, with $\rho_{\text{Cu}} \approx 1.7 \times 10^{-8} \Omega\cdot\text{m}$.

Electromotive Force (emf) and Internal Resistance

Concepts

• Electromotive Force (emf, $\mathcal{E}$): The energy provided per unit charge by a source (e.g., battery). Measured in volts (V).

• Internal Resistance (r): Real sources of emf have internal resistance, modeled as a resistor in series with an ideal emf.

• Terminal Voltage: The voltage across the terminals of a real battery is $V_{ab} = \mathcal{E} - Ir$.

Example: For a battery with $\mathcal{E} = 12\,\text{V}$ and $r = 2\,\Omega$ supplying $I = 2\,\text{A}$, $V_{ab} = 12\,\text{V} - (2\,\text{A})(2\,\Omega) = 8\,\text{V}$.

Electric Power in Circuits

Power Delivered to Circuit Elements

• Power (P): The rate at which energy is delivered or consumed in a circuit element.

• Given by $P = V_{ab} I$.

• For a resistor: $P = I^2 R = \frac{V_{ab}^2}{R}$.

Example: A $2\,\Omega$ resistor with $I = 2\,\text{A}$ dissipates $P = (2\,\text{A})^2 \times 2\,\Omega = 8\,\text{W}$.

Measuring Current and Voltage

Instrument Placement

• Ammeter: Measures current; must be placed in series with the circuit element.

• Voltmeter: Measures voltage; must be placed in parallel with the circuit element.

Example: To measure the current through and voltage across a resistor, place the ammeter in series and the voltmeter in parallel with the resistor.

Resistor Networks: Series and Parallel

Series Circuits

• Resistors are in series if connected end-to-end; the same current flows through each.

• Total resistance: $R_{\text{eq}} = R_1 + R_2 + \cdots + R_n$.

• Voltage divides among resistors: $V_{ab} = V_1 + V_2 + \cdots + V_n$.

Example: Three resistors of $2\,\Omega$, $3\,\Omega$, and $5\,\Omega$ in series have $R_{\text{eq}} = 10\,\Omega$.

Parallel Circuits

• Resistors are in parallel if connected across the same two points; the same voltage is across each.

• Total resistance: $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}$.

• Current divides among resistors.

Example: Two resistors of $6\,\Omega$ and $3\,\Omega$ in parallel: $\frac{1}{R_{\text{eq}}} = \frac{1}{6} + \frac{1}{3} = \frac{1}{2}$, so $R_{\text{eq}} = 2\,\Omega$.

Kirchhoff’s Laws

Junction (Current) Law

• The sum of currents entering a junction equals the sum leaving: $\sum I_{\text{in}} = \sum I_{\text{out}}$.

• Expresses conservation of charge.

Loop (Voltage) Law

• The sum of potential differences around any closed loop is zero: $\sum V = 0$.

• Expresses conservation of energy.

• Sign conventions: traversing a resistor in the direction of current, $-IR$; against current, $+IR$.

Example: In a loop with a battery and resistors, apply the loop law to solve for unknown currents.

Capacitors in Circuits (RC Circuits)

Charging a Capacitor

• When a capacitor is connected to a battery through a resistor, it charges over time.

• Charge on the capacitor: $q(t) = Q_f (1 - e^{-t/RC})$, where $Q_f = C\mathcal{E}$.

• Current during charging: $I(t) = \frac{\mathcal{E}}{R} e^{-t/RC}$.

• Time constant: $\tau = RC$ (characteristic time for charging/discharging).

Discharging a Capacitor

• When disconnected from the battery, the capacitor discharges through the resistor.

• Charge: $q(t) = Q_0 e^{-t/RC}$, where $Q_0$ is the initial charge.

• Current: $I(t) = -\frac{Q_0}{RC} e^{-t/RC}$ (negative sign indicates direction).

Energy in RC Circuits

• Total energy supplied by the battery: $U = C\mathcal{E}^2$.

• Half is stored in the capacitor, half dissipated as heat in the resistor during charging.

Summary Table: Series vs. Parallel Circuits

Property	Series	Parallel
Current	Same through all elements	Divides among branches
Voltage	Divides among elements	Same across all branches
Equivalent Resistance	$R_{\text{eq}} = \sum R_i$	$\frac{1}{R_{\text{eq}}} = \sum \frac{1}{R_i}$
Failure Effect	One fails, all fail	One fails, others work

Key Formulas

• Ohm's Law: $V = IR$

• Resistivity: $R = \rho \frac{l}{A}$

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

• Series Resistance: $R_{\text{eq}} = R_1 + R_2 + \cdots$

• Parallel Resistance: $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$

• Kirchhoff's Laws: $\sum I_{\text{in}} = \sum I_{\text{out}}$, $\sum V = 0$ (loop)

• RC Charging: $q(t) = Q_f (1 - e^{-t/RC})$

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

Additional info: Some diagrams and equations were inferred from standard physics curriculum to clarify the concepts and ensure completeness.