Current Electricity Study Notes

Electric Current ( $I$ ): Net movement of charge across a cross-sectional area per unit time: $I = \lim_{\Delta t \rightarrow 0} \frac{\Delta Q}{\Delta t}$ The SI unit of current is the ampere ( $\text{A}$ ).
Ohm's Law: Potential difference ( $V$ ) across a conductor is directly proportional to current ( $I$ ) flowing through it: $V = I R$ where $R$ is resistance, measured in ohms ( $\Omega$ ).
Resistance Geometry: Resistance depends on material resistivity ( $\rho$ ), length ( $l$ ), and cross-sectional area ( $A$ ): $R = \rho \frac{l}{A}$
Current Density ( $\mathbf{j}$ ) and Vector Ohm's Law: Current per unit area normal to flow: $\mathbf{j} = \sigma \mathbf{E}$ where $\sigma = \frac{1}{\rho}$ is electrical conductivity.

Thermal Motion: In the absence of an electric field, free electron motion is completely random due to collisions with fixed ions, yielding zero average thermal velocity.
Drift Velocity ( $\mathbf{v}_d$ ): Average velocity acquired by electrons due to an applied electric field $\mathbf{E}$ : $\mathbf{v}_d = -\frac{e \mathbf{E}}{m} \tau$ where $e$ is electron charge, $m$ is electron mass, and $\tau$ is relaxation time (average time between successive collisions).
Microscopic Current Relation: $I = n e A |\mathbf{v}_d|$ where $n$ is free electron number density.
Microscopic Conductivity and Resistivity: $\sigma = \frac{n e^2 \tau}{m} \quad \text{and} \quad \rho = \frac{m}{n e^2 \tau}$
Mobility ( $\mu$ ): Magnitude of drift velocity per unit electric field: $\mu = \frac{|\mathbf{v}_d|}{E} = \frac{e \tau}{m}$ SI unit of mobility is $\text{m}^2\,\text{V}^{-1}\,\text{s}^{-1}$ .

Limitations of Ohm's Law:
- Non-linear relation between $V$ and $I$ at high currents.
- Current magnitude changes when voltage sign is reversed (e.g., semiconductor diode).
- Non-unique relationship between $V$ and $I$ (e.g., gallium arsenide, $\text{GaAs}$ ).
Temperature Dependence of Resistivity: $\rho_T = \rho_0 [1 + \alpha (T - T_0)]$ where $\alpha$ is the temperature coefficient of resistivity ( $\text{K}^{-1}$ or $^\circ\text{C}^{-1}$ ).
Material Variations:
- Metals: Positive $\alpha$ ; higher temperatures cause more frequent collisions, reducing $\tau$ and increasing $\rho$
- Resistive Alloys (Nichrome, Manganin, Constantan): Very weak temperature dependence of resistivity.
- Semiconductors: Negative $\alpha$ ; higher temperatures significantly increase carrier density $n$ , overriding the decrease in $\tau$ and decreasing $\rho$

Electrical Power Loss: Rate of energy dissipated as heat in a conductor ("ohmic loss"): $P = I V = I^2 R = \frac{V^2}{R}$
Power Transmission: Cable power loss $P_c = \frac{P^2 R_c}{V^2}$ is minimized by transmitting power at high voltage $V$
Electromotive Force ( $\varepsilon$ ): Potential difference between positive and negative terminals of a cell in an open circuit ( $I = 0$ ): $\varepsilon = V_+ + V_-$
Terminal Voltage and Internal Resistance ( $r$ ): $V = \varepsilon - I r$ $I = \frac{\varepsilon}{R + r}$

Series Combination of Cells: $\varepsilon_{eq} = \varepsilon_1 + \varepsilon_2 \quad \text{and} \quad r_{eq} = r_1 + r_2$
Parallel Combination of Cells: $\frac{1}{r_{eq}} = \frac{1}{r_1} + \frac{1}{r_2} \quad \text{and} \quad \frac{\varepsilon_{eq}}{r_{eq}} = \frac{\varepsilon_1}{r_1} + \frac{\varepsilon_2}{r_2}$
Kirchhoff's Rules:
- Junction Rule: Total current entering a junction equals total current leaving it (conservation of charge).
- Loop Rule: Algebraic sum of potential changes around any closed loop is zero (conservation of energy).
Wheatstone Bridge: Four-resistor network ( $R_1, R_2, R_3, R_4$ ) used to determine an unknown resistance.
- Balance Condition: Zero current through the galvanometer ( $I_g = 0$ ) yields: $\frac{R_2}{R_1} = \frac{R_4}{R_3}$