A cell of emf ε and internal resistance r is connected to an external resistance R. The maximum power delivered to R is

1. Current in the loop: I = ε / (R + r). 2. Power delivered to R: P = I² R = ε² R / (R + r)². 3. Differentiate with respect to R and set dP/dR = 0. 4. This gives R = r, and substituting back yields P_max = ε² / (4 r). _Source: syllabus Physics, Ch 3 "Current Electricity", §3 EMF and internal resistance_