Current, Resistance, and Resistivity

Source lecture(s): SC134 Lec8

Intuition

Materials impede charge flow; resistivity is an intrinsic property, resistance depends on geometry.

Formal Definition

Ohm's law: $V = IR$. Resistivity: $\rho = RA/L$ Conductivity: $\sigma = 1/\rho$

Mathematical Formulation

\[R = \rho\frac{L}{A}$$ Power: $$P = IV = I^2R = V^2/R\]

Derivation

Drude model: $v_d = \mu E$, $J = \sigma E$, so $V = EL = (\rho L/A)I$.

Worked Example

1 km of 1 mm$^2$ copper ($\rho=1.7\times10^{-8}\,\Omega\cdot\text{m}$): $R = \rho L/A = 1.7\times10^{-8}\times 10^3/10^{-6} = 17\,\Omega$ — why long, thin cables need thick copper.

Common Mistakes

Confusing series vs parallel combination rules.
Using copper resistivity for nichrome.

Quiz

Q1. If length doubles and area halves, what happens to R?

Answer

It quadruples.