Current, Resistance, and Electromotive Force (EMF): Study Notes

Current, Resistance, and Electromotive Force (EMF)

Electric Current

Electric current is the rate at which electric charge flows through a surface or conductor. It is a fundamental concept in understanding how electrical circuits operate.

• Definition: Electric current (I) is defined as the amount of charge (Q) passing through a cross-sectional area (A) per unit time (t).

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

• Direction: By convention, current direction is the direction of positive charge flow, even though in metals, electrons (negative charges) are the actual charge carriers.

Formulas:

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

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

Current and Drift Speed

In a conductor, current is carried by charge carriers (typically electrons) moving with an average drift velocity due to an applied electric field.

• Drift Velocity (vd): The average velocity at which charge carriers move through a conductor under the influence of an electric field.

• Number Density (n): Number of charge carriers per unit volume.

• Charge per Carrier (q): For electrons, q = -e.

• Current Formula: $I = nqv_dA$

Microscopic View of Current

On a microscopic level, electrons move randomly but acquire a net drift velocity in the presence of an electric field, resulting in a current.

• Electrons undergo frequent collisions with atoms, causing a zigzag path.

• The net motion (drift) is opposite to the electric field for electrons.

Current Density

Current density (J) is the current per unit area of cross-section in a conductor.

• Formula: $J = \frac{I}{A} = nqv_d$

• SI Unit: A/m2

• Direction: Same as the direction of positive charge flow and the electric field.

Ohm’s Law and Resistance

Ohm’s Law relates the current through a conductor to the voltage across it and its resistance. It is an empirical law valid for many materials (ohmic materials).

• Ohm’s Law: $V = IR$

• Resistance (R): Opposition to current flow, measured in ohms (Ω).

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

• Conductivity (σ): Inverse of resistivity, $\sigma = 1/\rho$.

Temperature Dependence of Resistance

The resistance of most conductors increases with temperature. For some materials, such as semiconductors, resistance decreases with temperature.

• Linear Approximation: $\rho = \rho_0 [1 + \alpha (T - T_0)]$

• Temperature Coefficient (α): Indicates how resistivity changes with temperature.

• Superconductors: Materials whose resistance drops to zero below a critical temperature (Tc).

Electromotive Force (EMF) and Internal Resistance

Electromotive force (emf, ε) is the energy provided per unit charge by a source such as a battery. Real sources have internal resistance (r), which reduces the terminal voltage when current flows.

• Terminal Voltage: $V = \varepsilon - Ir$

• Current in Circuit: $I = \frac{\varepsilon}{R + r}$

• Open Circuit: Terminal voltage equals emf when I = 0.

Electrical Power in Circuits

Electrical power is the rate at which energy is delivered to a component in a circuit, such as a resistor.

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

• Unit: Watt (W), where 1 W = 1 J/s.

• Energy Consumption: Electrical energy is often measured in kilowatt-hours (kWh).

Summary Table: Resistivities and Temperature Coefficients

The following table summarizes the resistivities and temperature coefficients for various materials, which are essential for selecting materials in electrical applications.

Material	Resistivity (Ω·m)	Temperature Coefficient (°C-1)
Silver	1.59 × 10-8	3.8 × 10-3
Copper	1.7 × 10-8	3.9 × 10-3
Gold	2.44 × 10-8	3.4 × 10-3
Aluminum	2.82 × 10-8	3.9 × 10-3
Iron	1.0 × 10-7	6.5 × 10-3
Lead	2.2 × 10-7	3.9 × 10-3
Carbon	3.5 × 10-5	-0.5 × 10-3
Silicon	6.4 × 102	-75 × 10-3
Glass	1010 to 1014	—

Color Coding for Resistors

Resistor values are often indicated by colored bands. Each color corresponds to a digit, multiplier, and tolerance.

Color	Number	Multiplier	Tolerance
Black	0	1
Brown	1	101
Red	2	102
Orange	3	103
Yellow	4	104
Green	5	105
Blue	6	106
Violet	7	107
Gray	8	108
White	9	109
Gold		10-1	5%
Silver		10-2	10%
Colorless			20%

Ohmic and Nonohmic Materials

Ohmic materials have a linear relationship between current and voltage, while nonohmic materials do not.

• Ohmic: $I \propto \Delta V$ (straight line on I-V graph)

• Nonohmic: Nonlinear I-V relationship (e.g., diodes)

Applications: Power Transmission

Electric power is transmitted at high voltages and low currents to minimize energy loss due to resistance in transmission lines.

Additional info: These notes provide a comprehensive overview of electric current, resistance, Ohm's law, temperature effects, EMF, and power in circuits, suitable for college-level physics students preparing for exams.