Electric Conduction in Metals

Electric conduction in metals is carried by free electrons. These electrons have thermal energy and move randomly with average speeds of $10^6\ \text{ms}^{-1}$. When a voltage is applied, an electric field $E$ forces them to drift slowly opposite the field with drift velocity $v_d \sim 10^{-5}\ \text{ms}^{-1}$.

Substituting equation (1.5) into (1.3):

$I = \frac{Ne}{t} = \frac{nAle}{t} \Rightarrow v_d = \frac{l}{t} \Rightarrow I = neAv_d$

\therefore v_d = \frac{I}{neA} \tag{1.6}

Where:

• $n$: number of free electrons per unit volume
• $N = nAl$: Total number of electrons in volume $Al$ \tag{1.5}

J = \frac{I}{A} \tag{1.7}

Since $I = neAv_d$,

J = nev_d \tag{1.8}

The electric field $E$ produces force $F_e = -eE$. Newton's second law gives:

$F = ma \Rightarrow a = -\frac{eE}{m}$

Drift velocity acquired during relaxation time $\tau$:

v_d = -\frac{eE}{m}\tau \tag{1.9}

Using $E = -\frac{V}{l}$, equation (1.6), and substituting into (1.9):

I = \left( \frac{ne^2A\tau}{m l} \right) V \tag{1.10}

V = \left( \frac{ml}{ne^2A\tau} \right) I = IR \tag{1.11}

$R$: resistance in Ohms ($\Omega$)

G = \frac{1}{R} \tag{1.12}

(Conductance in $\Omega^{-1}$)

From equations (1.10) and (1.11):

R = \frac{ml}{ne^2A\tau} \tag{1.13}

R \propto \frac{l}{A} \Rightarrow R = \rho \frac{l}{A} \tag{1.14}

Comparing with (1.13):

\rho = \frac{m}{ne^2\tau} \tag{1.15}

\sigma = \frac{1}{\rho} = \frac{E}{J} = \frac{V/l}{I/A} = \frac{VA}{Il} = \frac{1}{R} \frac{A}{l} \tag{1.16}

I = \frac{\sigma A}{l}V \Rightarrow J = \sigma E \tag{1.17}

Let $R_0$ be resistance at $\theta_0$, and $R_\theta$ at $\theta$. Then:

$\Delta R = R_\theta - R_0 \propto R_0 (\theta - \theta_0)$

$\Delta R = \alpha R_0 (\theta - \theta_0)$

$\alpha$ is the temperature coefficient of resistance.

$R_\theta = R_0 + \alpha R_0 (\theta - \theta_0)$

Table Resistivity of Selected Materials (at Room Temperature)

Material	Resistivity $\rho$ (Ω·m)
Silver	1.6 × 10⁻⁸
Copper	1.7 × 10⁻⁸
Aluminium	2.8 × 10⁻⁸
Iron	9.8 × 10⁻⁸
Constantan	4.9 × 10⁻⁷
Mercury	9.8 × 10⁻⁷
Germanium	4.6 × 10⁻¹
Silicon	1.09 × 10³
Tin	6.4 × 10⁻⁸
Fused Quartz	7.5 × 10¹⁷
Brass	8.0 × 10⁻⁸
Tungsten	5.6 × 10⁻⁸
Carbon (graphite)	(2.5–5) × 10⁻⁶
Nichrome	1.1 × 10⁻⁶

a. Temperature Dependence of Resistance

Resistance of a conductor typically increases with temperature. For a metallic conductor, the relationship is approximately linear:

$R = R_0(1 + \alpha \Delta T)$

• $R_0$: Resistance at reference temperature (usually 0°C)
• $\alpha$: Temperature coefficient of resistance
• $\Delta T = T - T_0$: Temperature change

In semiconductors, resistance decreases with increasing temperature due to increased carrier density.

b. Combination of Resistors

Series Combination

$R_{\text{eq}} = R_1 + R_2 + \dots + R_n$

Same current flows through all resistors; voltage divides.

Parallel Combination

$\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \dots + \frac{1}{R_n}$

Same voltage across each resistor; current divides.

c. Combination of Cells

Series Combination of Cells

$\mathcal{E}_{\text{eq}} = \mathcal{E}_1 + \mathcal{E}_2 + \dots + \mathcal{E}_n$

Internal resistances add: $r_{\text{eq}} = r_1 + r_2 + \dots + r_n$

Parallel Combination of Cells

For identical cells:

$\mathcal{E}_{\text{eq}} = \mathcal{E}, \quad r_{\text{eq}} = \frac{r}{n}$

d. Kirchhoff's Laws

Kirchhoff's Current Law (KCL)

At any junction, the total current entering equals the total current leaving:

$\sum I_{\text{in}} = \sum I_{\text{out}}$

Kirchhoff's Voltage Law (KVL)

The sum of EMFs and potential drops in any closed loop is zero:

$\sum \mathcal{E} - \sum IR = 0$

e. Complex Circuit Analysis (Loop and Node Methods)

Node Voltage Method

Assign voltages to nodes and apply KCL using Ohm's law:

$\frac{V_1 - V_2}{R} + \frac{V_1 - V_3}{R'} + \dots = 0$

Mesh Current Method

Assign loop currents and apply KVL around each mesh.

$I_1(R_1 + R_3) - I_2 R_3 = \mathcal{E}_1$

f. Thevenin's Theorem

Any linear two-terminal network can be replaced by an equivalent circuit with a single voltage source and a series resistor:

• Thevenin Voltage $V_{\text{th}}$: Open-circuit voltage
• Thevenin Resistance $R_{\text{th}}$: Resistance seen from terminals with sources replaced by internal resistances

$V_{\text{load}} = V_{\text{th}} \cdot \frac{R_{\text{load}}}{R_{\text{load}} + R_{\text{th}}}$

g. Norton's Theorem

Equivalent to Thevenin's theorem, but expressed as a current source in parallel with a resistor:

$I_{\text{N}} = \frac{V_{\text{th}}}{R_{\text{th}}}, \quad R_{\text{N}} = R_{\text{th}}$

h. Power in Circuits

Power dissipated by a resistor:

$P = I^2 R = \frac{V^2}{R} = VI$

i. Maximum Power Transfer Theorem

Maximum power is transferred to the load when:

$R_{\text{load}} = R_{\text{th}}$

Maximum power:

$P_{\text{max}} = \frac{V_{\text{th}}^2}{4 R_{\text{th}}}$

j. Energy Considerations

Energy supplied by a cell:

$W = Q \mathcal{E}$

Energy converted to heat in resistance:

$W = I^2 R t$

k. Superposition Theorem

In a linear network with multiple sources, the response (voltage/current) in any element is the algebraic sum of the responses caused by each source independently (others replaced by internal resistances).