Electric Current, Ohm's Law and Resistivity notes: ISC Class 12 Physics

1. Free electrons, drift and mobility

Current $I = \Delta q/\Delta t$ (A = C s⁻¹); conventional current is opposite to electron flow.
Free electrons move randomly (~10⁵ m s⁻¹) colliding with ions; relaxation time τ = mean time between collisions (~10⁻¹⁴ s).
In a field: acceleration $eE/m$ ; drift velocity $v_d = eE\tau/m$ (opposite to E), ~10⁻⁴ m s⁻¹. Mobility $\mu = v_d/E = e\tau/m$ (m² V⁻¹ s⁻¹).
$I = n e A v_d$ (n = free electrons per m³): all electrons within $v_d\Delta t$ of a section cross it in $\Delta t$ .
Lamps light instantly because the field is set up along the wire at nearly the speed of light.

Same random kicks in both; the field adds a steady drift opposite to E (checked).

Formulas

Current and drift
$I = n e A v_d$
Drift velocity
$v_d = \dfrac{eE\tau}{m}$

Key terms

Relaxation time: Average time between successive collisions of a free electron with the ions.
Drift velocity: Average velocity acquired by free electrons in a conductor in an electric field.
Mobility: Drift velocity per unit electric field, eτ/m.

2. Ohm’s law in microscopic form

Current density $J = I/A = n e v_d$ (A m⁻²). With $v_d = eE\tau/m$ : $J = \dfrac{n e^2\tau}{m}E = \sigma E$ .
Conductivity $\sigma = \dfrac{n e^2\tau}{m}$ ; resistivity $\rho = \dfrac{m}{n e^2\tau}$ .
For a wire, $E = V/l$ , $J = I/A$ give $V = IR$ with $R = \rho l/A$ : Ohm’s law (V ∝ I at constant temperature).

Same random kicks in both; the field adds a steady drift opposite to E (checked).

Formulas

Ohm’s law (vector form)
$\vec J = \sigma\vec E$
Resistivity
$\rho = \dfrac{m}{n e^2\tau}$

Key terms

Current density: Current per unit cross-sectional area, J = I/A.

3. Resistance, resistivity and V–I graphs

$R = \rho l/A$ ; conductance $G = 1/R$ (S); conductivity $\sigma = 1/\rho$ (S m⁻¹).
Stretching to $n$ times the length at constant volume: $A \to A/n$ , $R \to n^2R$ .
Ohmic: V–I graph a straight line through the origin (metals at constant temperature). Non-ohmic: curved (filament lamp), one-way (diode), negative slope regions, or different in reverse.

metal wire at constant temperature: ohmic (R = 4 Ω)
filament lamp: non-ohmic, R rises as it heats

Ohmic: straight line through the origin. Non-ohmic: curved (or not through the origin, or different in each direction).

Drawn to scale: twice the length, half the cross-section, four times the resistance (checked).

Formulas

Resistance of a wire
$R = \dfrac{\rho l}{A}$

Key terms

Resistivity: Resistance of a conductor of unit length and unit cross-section; a property of the material.
Ohmic conductor: One for which V is proportional to I at constant temperature.

4. Temperature dependence

Metals: $\rho_T = \rho_0[1 + \alpha(T - T_0)]$ , α > 0 (copper about 3.9 × 10⁻³ °C⁻¹): n constant, τ decreases as ions vibrate more.
Semiconductors: ρ decreases with temperature (negative α): n increases sharply, outweighing the fall in τ.
Alloys (manganin, constantan): very small α, high ρ — standard resistors; nichrome for heaters.

copper (metal): rises almost linearly
semiconductor: falls steeply

Metals: τ falls as ions vibrate more. Semiconductors: n rises enormously, outweighing the fall in τ.

Formulas

Temperature dependence
$R_T = R_0[1 + \alpha(T - T_0)]$

Key terms

Temperature coefficient of resistivity: Fractional change in resistivity per degree rise in temperature.

5. Electrical energy and power

$P = VI = I^2R = V^2/R$ ; energy $W = Pt = VIt$ ; Joule heating $H = I^2Rt$ .
Commercial unit: 1 kW h (one “unit”) = $3.6 \times 10^{6}$ J. Bill = total kW h × rate.
Rated appliance: $R = V_\text{rated}^2/P_\text{rated}$ ; on a different supply $P' = V'^2/R$ .
Long-distance transmission at high voltage reduces current and so the $I^2R$ loss.

metal wire at constant temperature: ohmic (R = 4 Ω)
filament lamp: non-ohmic, R rises as it heats

Ohmic: straight line through the origin. Non-ohmic: curved (or not through the origin, or different in each direction).

Formulas

Electrical power
$P = VI = I^2R = \dfrac{V^2}{R}$

Key terms

Kilowatt hour: Energy used by a 1 kW device in one hour, 3.6 × 10⁶ J.

Exam tips

Derive $I = n e A v_d$ and ρ = m/(ne²τ); convert mm² to m².
Stretching: constant volume, R ∝ l².
Temperature: explain through n and τ.
Power: choose the form using the common quantity; bills in kW h.

1. Free electrons, drift and mobility

Formulas

Key terms

2. Ohm’s law in microscopic form

Formulas

Key terms

3. Resistance, resistivity and V–I graphs

Formulas

Key terms

4. Temperature dependence

Formulas

Key terms

5. Electrical energy and power

Formulas

Key terms

Exam tips

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