Ohm's Law

In 2003, a transmission line in northern Ohio sagged into an overgrown tree and tripped offline. The same power rerouted through remaining lines, increasing current, increasing resistive heating, causing more lines to sag and trip. The cascade blacked out 50 million people across eight states and Canada. The relationship governing this cascade: V = IR.

Ohm's Law states that current through a conductor is directly proportional to voltage and inversely proportional to resistance: V = IR, or equivalently I = V/R. Combined with P = IV, it yields the critical transmission loss formula: P_loss = I²R. Power wasted as heat scales with the square of current. This quadratic relationship is why even small increases in current during contingencies can rapidly overheat lines.

The three forms (V = IR, I = V/R, R = V/I) let you calculate any electrical quantity from the other two. For grid engineering, P_loss = I²R is the most consequential: it dictates conductor sizing, voltage levels, and the economic case for high-voltage transmission.

Worked Example

A 150 km transmission line has a total resistance of 8 ohms and carries 2,000 A.

• Calculate power loss. P_loss = I²R = (2,000)² x 8 = 32,000,000 W = 32 MW.
• If one of two parallel lines trips, all current shifts to the remaining line: 4,000 A.

What are the new losses?

P_loss = (4,000)² x 8 = 128,000,000 W = 128 MW. Doubling the current quadrupled the losses, from 32 MW to 128 MW. This is why contingency events cascade: the extra heating can cause the surviving line to sag into trees, tripping it too.

Ohm's Law and the I²R loss formula explain why every grid decision about voltage, conductor material, and redundancy traces back to the relationship between current and resistance.

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