Terminal Potential Difference

When a cell drives current through a circuit, only part of its electromotive force (emf) appears across the external circuit as usable voltage — this is called the terminal potential difference. Understanding the distinction between emf and terminal potential difference is one of the most frequently tested concepts in board and competitive exams.

The work done per unit charge in the external circuit only: V = \[\frac {W_{ext}}{q}\]. It is what a voltmeter connected across the cell's terminals actually measures.

Step 1: A cell of emf E drives a small charge dq around the circuit, doing total work dW.

Step 2: The same charge dq flows through every part of the circuit (series flow), so the total energy is shared across each component: dW₁, dW₂, dW₃,…

Step 3: Summing the energy shares and dividing by charge gives potential differences across each part:

Step 4 (Key Result): If the circuit has only one cell and one external resistor, this reduces to:

where V is the terminal potential difference, and v_internal is the voltage drop inside the cell due to internal resistance.

Think of emf as your total monthly salary and terminal potential difference as your take-home pay after taxes (internal resistance). The "tax" (internal resistance drop) is deducted before the money reaches your usable spending (external circuit).