A battery will deliver its maximum power when its load resistance is?

• A. Zero
• B. Twice the internal resistance
• C. Half the internal resistance
• D. Equal to the internal resistance

Answer: (D)

When you want to pull the most power from a battery, the load should match the battery’s own resistance. Model the battery as an ideal voltage source E in series with an internal resistance r, and connect an external load R. The current is I = E/(R + r), and the power delivered to the load is P = I^2 R = E^2 R/(R + r)^2. To maximize P with respect to R, you find the point where dP/dR = 0, which gives R = r. At this point the load voltage is V_L = E/2 and the current is I = E/(2r), giving the maximum load power P_max = E^2/(4r). If the load is smaller than r, most of the voltage drops across the battery’s internal resistance; if larger, the current drops too much. Equaling them balances the voltage and current to achieve the greatest power transfer.