Current Electricity: Ohm's Law, Kirchhoff's & Circuits

Electric current is the rate of flow of charge: I = dQ/dt. SI unit: ampere (A). Dimensional formula: [A] (base quantity). Conventional current flows from higher to lower potential (opposite to electron flow). Current density J = I/A = nev_d, where n is the number density of free electrons, e is the electron charge, and $v_{d}$ is the drift velocity.

Drift Velocity: $v_{d}$ = eE*$\tau$/m, where E is the electric field, $\tau$ is the mean relaxation time, and m is the electron mass. Typical drift velocity is ~10^(-4) m/s, while thermal velocity is ~$10^{5}$ m/s. The signal (electric field) propagates at nearly the speed of light, but electrons drift very slowly.

Ohm's Law: V = IR, where V is the potential difference, I is the current, and R is the resistance.
In microscopic form: J = $\sigma$E, where $\sigma$ = $ne^{2}$$\tau$/m is the conductivity.
Resistance R = $\rho$*L/A, where $\rho$ = 1/$\sigma$ is the resistivity, L is the length, and A is the cross-sectional area. SI unit of R: ohm ($\omega$). Dimensions: [M $L^{2}$ T^(-3) A^(-2)].

Resistivity: $\rho$ = m/($ne^{2}$$\tau$). SI unit: ohm-meter. Depends on material and temperature, NOT on dimensions.
For metals: $\rho$ increases with temperature: $\rho$ = $\rho_{0}$(1 + $\alpha$$\delta_{T}$), where $\alpha$ is the temperature coefficient of resistance (~4 x 10^(-3) /K for most metals). For semiconductors and insulators: $\rho$ decreases with increasing temperature.

Combinations of Resistors:

• Series: $R_{eq}$ = R₁ + R₂ + ... (current same, voltage adds)
• Parallel: 1/$R_{eq}$ = 1/R₁ + 1/R₂ + ... (voltage same, current adds)
• Two in parallel: $R_{eq}$ = R₁*R₂/(R₁+R₂)

Kirchhoff's Laws:

1. Junction Rule (KCL): Sum of currents entering a junction = sum leaving. Based on conservation of charge: sum(I) = 0 at any node.
2. Loop Rule (KVL): Sum of potential changes around any closed loop = 0. Based on conservation of energy. Convention: voltage rises (through battery from - to +) are positive; voltage drops (through resistor in current direction) are negative.

EMF and Internal Resistance: A real battery has emf ($\epsilon$) and internal resistance r.
Terminal voltage: V = $\epsilon$ - Ir (discharging) or V = $\epsilon$ + Ir (charging).
Maximum current (short circuit): $I_{max}$ = $\epsilon$/r.

Wheatstone Bridge: Balanced when P/Q = R/S (no current through galvanometer). Used for precise resistance measurement. In balanced condition, the bridge can be simplified by removing the galvanometer arm.

Meter Bridge (Slide Wire Bridge): Based on Wheatstone bridge principle. Unknown resistance X = R * l/(100 - l), where R is the known resistance and l is the balancing length.

Potentiometer: Measures EMF without drawing current. Principle: V proportional to length l when constant current flows through uniform wire.
Comparison of EMFs: $\frac{\epsilon_{1}}{\epsilon_{2}}$ = $\frac{l_{1}}{l_{2}}$.
Internal resistance measurement: r = R($l_{1}$ - $l_{2}$)/$l_{2}$, where $l_{1}$ is the balance length with open circuit and $l_{2}$ with resistance R across the cell.

Power: P = VI = $I^{2}$R = $V^{2}$/R.
Maximum power transfer occurs when external resistance equals internal resistance: R = r (gives $P_{max}$ = $\epsilon$²/(4r)).

Heating Effect: H = $I^{2}$Rt (Joule's law). In series: H proportional to R (same I). In parallel: H proportional to 1/R (same V).