Current Electricity

Ohm’s law says the potential difference across a conductor is proportional to the current through it, at constant temperature:

For a real cell, the EMF $\varepsilon$ is what it would deliver with no current flowing; once current flows, some voltage is lost across its own internal resistance $r$:

For a complete circuit with external resistance $R$, solving for the current gives:

Power delivered to the external resistor is $P = I^2R$. Substituting $I$ and maximising over $R$ (by calculus, or by symmetry of the resulting curve) gives the maximum power transfer result:

The entire chapter hinges on a distinction that’s easy to blur under exam pressure: current and power don’t move together once you change the resistance they’re flowing through.

Shrinking R: current $I=\varepsilon/(R+r)$ rises monotonically as $R$ falls, all the way to its maximum at $R=0$ (a short circuit). But power to $R$, $P=I^2R$, is the product of a rising $I^2$ and a falling $R$. Near $R=0$, the falling $R$ wins, and $P\to0$ even as $I$ approaches its peak.

Growing R: at the other extreme, $I\to0$ as $R\to\infty$, so $P=I^2R\to0$ here too, despite $R$ itself being huge.

Between these two zeros, $P(R)$ must have a maximum, and it sits exactly at $R=r$. Students who reason “smaller resistance always means more delivered power” get this wrong on reflex, because they’re tracking $I$ alone and forgetting it’s $I^2R$ that matters.