Electric Current, Resistance, and Power: Study Notes

Electric Current, Resistance, and Power

Electric Current

Electric current is the flow of electric charge through a conductor or medium. It is a fundamental concept in electricity and is measured in amperes (A).

• Definition: The rate of charge crossing an area per unit time.

• Unit: 1 ampere (A) = 1 coulomb/second (C/s).

• Average current: $I = \frac{\Delta Q}{\Delta t}$

• Instantaneous current: $I = \frac{dQ}{dt}$

• Direction: By convention, current direction is the direction of positive charge flow. For negative charges, current is in the opposite direction of their motion.

Example: If 2 C of charge pass through a wire in 4 seconds, the average current is $I = \frac{2\ \mathrm{C}}{4\ \mathrm{s}} = 0.5\ \mathrm{A}$.

Direction of Current

The direction of current depends on the type and direction of moving charges:

• Positive charges moving right: Current to the right.

• Negative charges moving right: Current to the left.

• Equal amounts of positive and negative charges moving in opposite directions: Net current is the sum of both contributions.

• No net movement of charge: No current.

Example: In a solution, if positive ions move left and negative ions move right in equal amounts, the net current is to the left.

Microscopic Current

Microscopic current describes current at the level of individual charge carriers in a conductor.

• Depends on:

  • n: Number of charge carriers per unit volume

  • q: Charge of each carrier

  • vd: Average drift velocity of carriers (proportional to electric field $E$)

  • A: Cross-sectional area

• Formula: $I = n A q v_d$

Example: In a copper wire, electrons (negative charge carriers) move with a small drift velocity opposite to the electric field, resulting in a current in the direction of the field.

Current Density

Current density is the amount of current flowing per unit area of a conductor.

• Definition: $\vec{J} = \frac{I}{A}$ (if uniform and perpendicular to the surface)

• Vector quantity: Direction is the same as current for positive charges, opposite for negative charges.

• Units: A/m2

• General case: $I = \int \vec{J} \cdot d\vec{A}$

• Microscopic relation: $\vec{J} = n q \vec{v}_d$

Example: If a current of 2 A flows through a wire of area 0.5 mm2, $J = \frac{2}{0.5 \times 10^{-6}} = 4 \times 10^6$ A/m2.

Example: Current Density in Ion Channels

Ion channels in cell membranes allow ions to pass through, creating a current. For a channel with a circular cross-section of radius $1.5 \times 10^{-9}$ m, passing $1.1 \times 10^4$ potassium ions in 1 ms:

• Total charge: $Q = 1.1 \times 10^4 \times 1.6 \times 10^{-19} = 1.8 \times 10^{-15}$ C

• Current: $I = \frac{1.8 \times 10^{-15}}{1 \times 10^{-3}} = 1.8 \times 10^{-12}$ A = 1.8 pA

• Area: $A = \pi r^2 = \pi (1.5 \times 10^{-9})^2 = 7.07 \times 10^{-18}$ m2

• Current density: $J = \frac{I}{A} = \frac{1.8 \times 10^{-12}}{7.07 \times 10^{-18}} = 2.5 \times 10^5$ A/m2

Conduction in Metals

Metals are good conductors because they contain many free electrons that move randomly at high speeds. When an electric field is applied, these electrons acquire a small net drift velocity opposite to the field, resulting in a current.

• Random motion: No net current without an electric field.

• Drift velocity: Small average velocity superposed on random motion due to the field.

Ohm's Law: Microscopic and Macroscopic Forms

Ohm's law relates current, voltage, and resistance in conductors. It can be expressed at both the microscopic (local) and macroscopic (circuit) levels.

• Microscopic form: $\vec{J} = \sigma \vec{E}$, where $\sigma$ is conductivity.

• Alternative form: $\vec{J} = \frac{\vec{E}}{\rho}$, where $\rho$ is resistivity.

• Macroscopic form: $V = IR$

• Resistance: $R = \rho \frac{L}{A}$, where $L$ is length and $A$ is cross-sectional area.

• Conductivity and resistivity: $\sigma = \frac{1}{\rho}$

Example: A copper wire of length 2 m and area 1 mm2 ($\rho = 1.7 \times 10^{-8}$ Ω·m): $R = 1.7 \times 10^{-8} \times \frac{2}{1 \times 10^{-6}} = 0.034$ Ω.

Power in Electric Circuits

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

• Definition: $P = IV$

• Alternative forms:

  • $P = I^2 R$

  • $P = \frac{V^2}{R}$

• Interpretation: Power is dissipated as heat in resistors or converted to other forms of energy (mechanical, light, etc.).

Example: A 2 Ω resistor with 3 A current: $P = (3)^2 \times 2 = 18$ W.

Conductors, Insulators, and Semiconductors

Materials are classified by their ability to conduct electric current:

• Conductors: Low resistivity, high conductivity (e.g., metals).

• Insulators: High resistivity, low conductivity (e.g., glass, rubber).

• Semiconductors: Intermediate properties; conductivity can be modified (e.g., silicon).

Summary Table: Key Equations

Quantity	Equation
Resistance	$R = \rho \frac{L}{A}$
Current (average)	$I = \frac{\Delta Q}{\Delta t}$
Current (microscopic)	$I = n A q v_d$
Current density	$\vec{J} = n q \vec{v}_d$
Ohm's Law (macroscopic)	$V = IR$
Ohm's Law (microscopic)	$\vec{J} = \sigma \vec{E}$
Power	$P = IV$, $P = I^2 R$, $P = \frac{V^2}{R}$

Applications and Safety

• Biological currents: Electric currents in the body are essential for nerve and muscle function (e.g., EEG, EKG).

• Safety: Currents above 15 mA can be dangerous or fatal. Effects range from sensation to cardiac arrest depending on magnitude.

Additional info: The notes include both conceptual explanations and worked examples, as well as diagrams illustrating current direction and microscopic current. The summary table consolidates key equations for quick reference.