Electric Current, Resistance, and Energy in Circuits: Series and Parallel Elements

Electric Current and Ohm's Law

Definition and Properties of Electric Current

Electric current is a fundamental concept in circuit analysis, describing the flow of electric charge through a conductor. When a potential difference (voltage) is applied across a conducting wire, charges move to reduce the electric field inside. If the voltage is maintained, charges continue to flow, supplied by a battery or other source.

• Electric Current (I): The net flow of charge per unit time through a cross-sectional area. SI unit: ampere (A), where 1 A = 1 coulomb/second.

• Ohm's Law: Relates current, voltage, and resistance in a circuit. The empirical relationship is given by: $V = IR$ Alternatively, emphasizing resistance as the current-limiting factor: $I = \frac{V}{R}$

• Resistance (R): The property of a material and its geometry that opposes current flow. It is determined by the material's resistivity and the physical dimensions of the conductor:

• $R = \frac{\rho L}{A}$ where $\rho$ is the resistivity (material property), $L$ is the length, and $A$ is the cross-sectional area.

Resistors in Circuits: Series and Parallel Configurations

Resistors in Series

Resistors are in series when the same current flows sequentially through each resistor. The total voltage drop across all resistors equals the battery voltage.

• Current: Same through each resistor.

• Voltage Drop: Each resistor has a voltage drop given by Ohm's Law.

• Total Resistance: The equivalent resistance is the sum of individual resistances: $R_{eq} = R_1 + R_2 + \ldots$

• Example: If $R_1 = 2\,\Omega$ and $R_2 = 3\,\Omega$, then $R_{eq} = 5\,\Omega$.

Resistors in Parallel

Resistors are in parallel when each resistor is connected across the same two points, providing multiple paths for current. The voltage across each resistor is the same.

• Voltage: Same across each resistor.

• Current: Divided among the parallel branches; total current is the sum of currents through each resistor.

• Total Resistance: The equivalent resistance is found by the reciprocal sum: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$

• Example: If $R_1 = 2\,\Omega$ and $R_2 = 3\,\Omega$, then $\frac{1}{R_{eq}} = \frac{1}{2} + \frac{1}{3} = \frac{5}{6}$, so $R_{eq} = 1.2\,\Omega$.

Comparison: Series vs. Parallel

The formulas for combining resistors in series and parallel are opposite to those for capacitors. In series, resistances add; in parallel, their reciprocals add.

Configuration	Resistors	Capacitors
Series	$R_{eq} = R_1 + R_2 + \ldots$	$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots$
Parallel	$\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$	$C_{eq} = C_1 + C_2 + \ldots$

Energy and Power in Electric Circuits

Energy Transfer and Dissipation

As charges move from higher to lower potential, potential energy is converted to kinetic energy. In resistive wires, collisions cause energy to dissipate as heat.

• Change in Energy: $dU = -dQ V$

• Power Dissipation: The rate at which electrical energy is converted to heat: $P = IV$ Using Ohm's Law, alternative forms: $P = I^2 R$ $P = \frac{V^2}{R}$

• Units: 1 Watt (W) = 1 Joule/second (J/s)

• Example: A light bulb with $R = 60\,\Omega$ and $V = 120\,\text{V}$ dissipates $P = \frac{120^2}{60} = 240\,\text{W}$.

Applications: Light Bulbs and Circuit Puzzlers

Light Bulbs as Resistors

Light bulbs are modeled as resistors. Their brightness depends on the power dissipated, which is determined by the current and resistance.

• Brightness: Proportional to power $P = IV = I^2 R$.

• More Current: Higher current through the bulb means greater brightness.

Common Circuit Puzzlers

• Puzzler #1: In series, all current flows through all resistors; no alternate path.

• Puzzler #2: In parallel, the voltage across each resistor is the same. As more resistors are added, total current increases and equivalent resistance decreases.

• Puzzler #3: Comparing circuits, the one with more parallel paths has higher total current.

• Puzzler #4: A wire with very low resistance diverts all current through it, bypassing other resistors.

• Puzzler #5: Closing a switch can bypass a bulb, decreasing total resistance and increasing current; the remaining bulb gets brighter.

• Puzzler #6: Current divides among branches according to their resistance; more current flows through the branch with less resistance.

Additional info: The notes reference drawing circuit diagrams for clarity. In exam settings, always sketch series and parallel arrangements to visualize current paths and voltage drops.