The Mathematical Derivation of Ohm's Law and Joule's Power Law

Nearly all circuit analysis begins with two universal relationships: Ohm's Law ($V = IR$) and Joule's Power Law ($P = I^2 R = VI$). While commonly accepted as empirical axioms, these equations represent rigorous consequences of electrodynamics, conservation of energy, and microscopic statistical mechanics in crystalline metallic conductors.

1. Microscopic Derivation of Ohm's Law (The Drude Model)

Formulated by Paul Drude in 1900, the classical electron gas model treats conduction electrons inside a metal as non-interacting particles moving among fixed positive metallic cations. When an electric field $ec{E}$ is applied across a conductor of length $L$, the potential difference is:

Each electron experiences an electrostatic Lorentz force $F = -eE$. By Newton's Second Law, the acceleration between lattice collisions is:

Electrons repeatedly collide with vibrating lattice ions, completely randomizing their momentum. If $ au$ is the mean relaxation time between successive collisions (typically $sim 10^{-14}$ seconds in copper at room temperature), the net steady-state drift velocity is:

Electric current density $J$ (current per unit area $A$) is given by the charge density multiplied by drift velocity:

Defining electrical conductivity $sigma$ as $sigma = rac{n e^2 au}{m_e}$, this yields Ohm's Law in microscopic point form:

Now integrate over a macroscopic wire of length $L$ and cross-sectional area $A$:

2. Derivation of Joule's Power Law from Conservation of Energy

James Prescott Joule demonstrated in 1841 that mechanical work converts identically into thermal heat. Consider an infinitesimal charge element $dq$ passing through an electrical potential difference $V$:

Power $P$ is by definition the time rate of doing work ($P = rac{dW}{dt}$):

Now substitute Ohm's Law ($V = IR$ and $I = V/R$):

• Substituting $V = IR$: $P = (I cdot R) cdot I = mathbf{I^2 R}$
• Substituting $I = V / R$: $P = V cdot (V / R) = mathbf{V^2 / R}$

3. Grid Implication: Why the World Transmits at Ultra-High Voltage

The equation $P_{ ext{loss}} = I^2 R$ explains why national electric grids transmit bulk electricity at hundreds of thousands of volts (e.g. 500 kV) rather than utilization voltage (120V or 240V).

To deliver power $P = V cdot I$, increasing transmission voltage by a factor of 100 reduces line current by a factor of 100 ($I = P/V$). Because conductor line loss scales with the square of current ($I^2$):