Electric Current, Ohm's Law, and Electrical Circuits

Electric Current and Power

Definition of Electric Current

Electric current is the rate at which electric charge flows through a conductor. It is a fundamental concept in the study of electricity and circuits.

• Current (I): The amount of charge (Q) passing through a cross-section per unit time (t).

• Formula:

$I = \frac{\Delta Q}{\Delta t}$

• SI Unit: Ampere (A), where 1 A = 1 C/s.

Current Direction and Types

• Conventional Current: Direction in which positive charges would flow (from higher to lower potential).

• Electron Flow: Actual flow of electrons is opposite to the direction of conventional current.

Electric Power in Circuits

Electric power is the rate at which energy is transferred or converted by an electric circuit.

• Formula:

$P = IV$

• Where P is power (watts), I is current (amperes), and V is voltage (volts).

• Alternative forms using Ohm's Law:

$P = I^2R$

$P = \frac{V^2}{R}$

• Example: If a current of 2 A passes through a resistor with a voltage of 5 V, the power dissipated is $P = (2\,\text{A})(5\,\text{V}) = 10\,\text{W}$.

Resistance and Ohm's Law

Ohm's Law

Ohm's Law relates the voltage across a conductor to the current flowing through it and its resistance.

• Formula:

$V = IR$

• Where V is voltage (volts), I is current (amperes), and R is resistance (ohms, Ω).

• Applies to ohmic materials (materials with constant resistance).

Resistance

• Definition: Resistance is a measure of how much a material opposes the flow of electric current.

• Formula for a uniform wire:

$R = \rho \frac{L}{A}$

• Where ρ is resistivity (Ω·m), L is length (m), and A is cross-sectional area (m²).

• Resistivity (ρ): A material property that quantifies how strongly a material opposes current flow.

Example Calculation

• Given a copper wire of length 2 m and cross-sectional area 1 mm² ($1 \times 10^{-6}$ m²), with resistivity $1.7 \times 10^{-8}$ Ω·m:

$R = (1.7 \times 10^{-8}) \frac{2}{1 \times 10^{-6}} = 0.034\,\Omega$

Resistors in Circuits

Series and Parallel Combinations

Resistors can be combined in series or parallel to achieve desired resistance values in circuits.

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

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

• Key Point: In series, current is the same through all resistors; in parallel, voltage is the same across all resistors.

Example Table: Series vs. Parallel Resistors

Configuration	Equivalent Resistance	Current	Voltage
Series	$R_{\text{eq}} = R_1 + R_2 + \ldots$	Same through all	Divided among resistors
Parallel	$\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$	Divided among branches	Same across all

Microscopic View: Drift Velocity and Current Density

Drift Velocity

Drift velocity is the average velocity of charge carriers (usually electrons) due to an electric field in a conductor.

• Formula:

$I = n q v_d A$

• Where n is the number density of charge carriers, q is the charge of each carrier, v_d is drift velocity, and A is cross-sectional area.

Current Density

• Definition: Current density (J) is the current per unit area.

• Formula:

$J = \frac{I}{A}$

• Also, $J = n q v_d$

• SI Unit: A/m²

Conductivity and Resistivity

Definitions

• Conductivity (σ): A measure of a material's ability to conduct electric current.

• Resistivity (ρ): The reciprocal of conductivity; a measure of how strongly a material opposes current flow.

• Relationship: $\sigma = \frac{1}{\rho}$

Temperature Dependence

• For most conductors, resistivity increases with temperature.

• Approximate relationship:

$\rho = \rho_0 [1 + \alpha (T - T_0)]$

• Where ρ₀ is the resistivity at reference temperature T₀, and α is the temperature coefficient of resistivity.

Summary Table: Key Electrical Quantities

Quantity	Symbol	Unit	Formula
Current	I	Ampere (A)	$I = \frac{\Delta Q}{\Delta t}$
Voltage	V	Volt (V)	$V = IR$
Resistance	R	Ohm (Ω)	$R = \rho \frac{L}{A}$
Power	P	Watt (W)	$P = IV$
Current Density	J	A/m²	$J = \frac{I}{A}$
Conductivity	σ	S/m	$\sigma = \frac{1}{\rho}$

Additional info: Some context and definitions have been expanded for clarity and completeness, as the original notes were brief and used shorthand notation.