Current Electricity

Current is the rate of flow of charge. SI unit: ampere (A = C s⁻¹). Conventional current flows from high to low potential; electrons flow oppositely.

Current density J is the current per unit cross-sectional area. It is a vector in the direction of conventional current flow. SI unit: A m⁻².

Average velocity acquired by electrons in a conductor under field E. τ is the mean relaxation time, m is electron mass, e is electron charge. Typically ~10⁻⁴ m/s — far slower than random thermal speeds.

Current in terms of carrier density n, charge e, cross-sectional area A, and drift velocity vd. Fundamental relation connecting microscopic motion to macroscopic current.

For ohmic conductors, voltage is proportional to current at constant temperature. R is the resistance. Ohm's law is not a universal law — it holds only for materials where R is independent of V and I.

Resistance depends on geometry (L, A) and material property ρ (resistivity). σ is conductivity. SI unit of ρ: Ω·m. SI unit of σ: S m⁻¹ (siemens per metre).

For metals, resistance increases linearly with temperature. α is the temperature coefficient of resistance (unit: K⁻¹). For semiconductors, α is negative — resistance decreases with temperature.

Resistivity in terms of electron mass m, carrier density n, charge e, and mean relaxation time τ. Directly derivable from the drift velocity expression and Ohm's law.

Same current through each resistor; voltages add. Equivalent resistance is the sum. R_eq is always greater than the largest individual resistance.

Same voltage across each resistor; currents add. For two resistors: R_eq = R₁R₂/(R₁+R₂). R_eq is always less than the smallest individual resistance.

Terminal voltage V equals EMF ε minus voltage drop across internal resistance r during discharge. During charging, V = ε + Ir — external source must exceed EMF.

For n identical cells in series: ε_eq = nε, r_eq = nr. Series combination increases EMF and internal resistance equally — beneficial when external resistance is much larger than r.

For n identical cells in parallel: EMF stays the same, internal resistance reduces to r/n. Parallel combination is beneficial when external resistance is much smaller than r.

Algebraic sum of currents at any junction is zero. Equivalently: sum of currents entering a junction equals sum leaving. Statement of charge conservation.

Algebraic sum of potential differences around any closed loop is zero. Statement of energy conservation. Sign convention: potential rises across EMF sources, drops across resistors in the direction of current.

At balance (no current through galvanometer): P/Q = R/S. Used to find unknown resistance S when P, Q, R are known. Balance is independent of EMF and galvanometer resistance.

Practical form of Wheatstone bridge using a uniform wire of length 100 cm. l is the balance length from one end. Unknown resistance S = R(100 − l)/l.

Two EMFs are compared by finding their balance lengths l₁ and l₂ on the potentiometer wire. At balance, no current is drawn from the cell — gives true EMF, not terminal voltage.

Internal resistance of a cell measured using a potentiometer. l₁ is balance length with cell in open circuit, l₂ with external resistance R connected. Derived from EMF and terminal voltage comparison.

Rate of energy dissipation in a resistor. Three equivalent forms. SI unit: watt (W). For a source of EMF: P_delivered = εI, P_internal = I²r, P_external = I²R.

Heat produced in a resistor carrying current I for time t. H = Pt. In calories: H (cal) = I²Rt/4.18. Basis of all resistive heating devices.

Fraction of total power delivered to external circuit. Maximum power transfer (η = 50%) occurs when R = r, but maximum efficiency requires R ≫ r.

Maximum power delivered to external resistance R occurs when R equals internal resistance r. At this condition efficiency is 50% — half the total power is wasted internally.

Mobility is the drift velocity per unit electric field. SI unit: m² V⁻¹ s⁻¹. Relates to conductivity: σ = neμ. Higher mobility → better conductor at a given carrier density.