Electric Circuits: Fundamentals, Laws, and Applications

Electric Circuits and Circuit Elements

Definition of an Electric Circuit

An electric circuit is a closed loop network that provides a return path for the flow of electric current. In most practical circuits, the current flows in a constant direction, known as direct current (DC). In this context, circuits are typically assumed to be in a steady state, with constant magnitude and direction of current.

• Direct Current (DC): Electric current that flows in one direction only.

• Steady State: Condition where current and voltage remain constant over time.

Basic Circuit Elements

Circuit elements are the fundamental components used to build electric circuits. Each element has a specific symbol used in circuit diagrams.

• Battery: Provides electromotive force (emf) to drive current.

• Wire: Conducts electric current with negligible resistance.

• Resistor: Circuit element with significant resistance, used to control current.

• Bulb: Converts electrical energy into light and heat.

• Junction: Point where three or more conductors meet.

• Capacitor: Stores electric charge and energy.

• Switch: Opens or closes the circuit to control current flow.

Circuit Diagrams

Circuit diagrams use standardized symbols to represent circuit elements, making it easier to analyze and communicate circuit designs.

• The longer line in the battery symbol represents the positive terminal.

• The battery's emf is indicated beside the symbol.

• + and – symbols are often shown for clarity.

Electromotive Force (emf) and Batteries

Definition and Role of emf

To maintain a potential difference between the battery electrodes, work must be done to convert chemical energy into electrical energy. The electromotive force (emf) is defined as the work done per unit charge:

$\text{emf} = \frac{W_{\text{chem}}}{q}$

• Unit of emf: Volt (V)

• If a battery is not connected to a load, the potential difference between its terminals equals the emf.

• If a load is connected, some energy is lost inside the battery due to internal processes, so the terminal voltage is less than the emf.

Kirchhoff’s Laws

Kirchhoff’s Junction Law (Current Law)

At any junction in a circuit, the law of conservation of current requires that the total current entering the junction equals the total current leaving:

$\sum I_{\text{in}} = \sum I_{\text{out}}$

• This is a statement of conservation of charge.

• Also known as Kirchhoff’s Current Law (KCL).

Kirchhoff’s Loop Law (Voltage Law)

For any closed path (loop) in a circuit, the sum of all potential differences encountered is zero:

$\sum (\Delta V)_i = 0$

• This is a statement of conservation of energy.

• Also known as Kirchhoff’s Voltage Law (KVL).

Analyzing Simple Circuits

Battery-Resistor Circuits

In a simple circuit with a battery and a resistor:

• The current is constant throughout the circuit.

• In the ideal case, wires have zero resistance, so all voltage drop occurs across the resistor.

• Ohm’s Law applies: $V = IR$

Steps for Circuit Analysis

1. Draw the circuit diagram using standard symbols.

2. Assign a direction to the current (usually from positive to negative terminal of the battery).

3. Apply Kirchhoff’s laws and Ohm’s law to determine voltages and currents.

Resistors in Series and Parallel

Resistors in Series

When resistors are connected end-to-end, they are in series. The same current flows through each resistor, but the voltage divides among them.

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

• Total voltage: $V = V_1 + V_2 + \cdots + V_n$

• Current: $I = \frac{V}{R_{\text{eq}}}$

Resistors in Parallel

When resistors are connected across the same two points, they are in parallel. The voltage across each resistor is the same, but the current divides among them.

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

• Total current: $I = I_1 + I_2 + \cdots + I_n$

• Voltage: $V = V_1 = V_2 = \cdots = V_n$

Comparison Table: Series vs. Parallel

Property	Series	Parallel
Current	Same through all resistors	Splits among branches
Voltage	Divides among resistors	Same across all resistors
Equivalent Resistance	Sum: $R_{\text{eq}} = R_1 + R_2 + \cdots$	Reciprocal sum: $1/R_{\text{eq}} = 1/R_1 + 1/R_2 + \cdots$

Internal Resistance of Batteries

Concept of Internal Resistance

Real batteries have an internal resistance $r$ in series with the external load $R$. This affects the terminal voltage when current flows:

• Terminal voltage: $V = \text{emf} - Ir$

• Total resistance: $R_{\text{total}} = R + r$

• Current: $I = \frac{\text{emf}}{R + r}$

The terminal voltage decreases as the current increases, due to the voltage drop across the internal resistance.

Measuring Voltage and Current

Voltmeters and Ammeters

• Voltmeter: Measures potential difference; connected in parallel; has very high resistance to minimize current draw.

• Ammeter: Measures current; connected in series; has very low resistance to minimize voltage drop.

• Both analog and digital meters exist; analog meters are based on galvanometers.

Electrical Power Dissipation

Power in Resistors

When current flows through a resistor, electrical energy is converted into heat. The rate of energy dissipation (power) is given by:

• $P = IV$

• Using Ohm’s Law: $P = I^2R = \frac{V^2}{R}$

• Power increases with current and voltage, but the relationship with resistance depends on which variable is held constant.

Applications and Examples

• Lightbulbs: For a given voltage, a bulb with lower resistance draws more current and is brighter.

• Power transmission: To minimize power loss ($P_{\text{loss}} = I^2R$), high voltage and low current are used in transmission lines.

Example Table: Power and Resistance in Lightbulbs

Bulb Power (W)	Voltage (V)	Resistance (Ω)
60	120	240
100	120	144

Additional info: Resistance calculated as $R = V^2/P$ for each bulb.

Summary of Key Equations

• Ohm’s Law: $V = IR$

• 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: $\sum I_{\text{in}} = \sum I_{\text{out}}$, $\sum (\Delta V)_i = 0$

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

• Terminal Voltage: $V = \text{emf} - Ir$