Several identical batteries are connected in series to power a research-grade resistor whose resistance is RL. Each battery has emf E and internal resistance r. The number of batteries connected in series is N (where N≥1). Assume the internal resistance of the connecting wires is negligible.
(1) Derive an expression for the current I through the load resistor in terms of N, E, r, and RL. Show all algebraic steps.
(2) Using your result from part (a), obtain an expression for the terminal potential difference across the entire battery pack while the load is connected.
(3) Define the efficiency η of the battery pack as the ratio of useful power delivered to the load to the total power produced by the batteries. Derive an expression for η in terms of N, r, and RL (the emf cancels).
(4) For fixed E, r, and RL, determine whether adding more batteries in series increases, decreases, or leaves unchanged (i) the current and (ii) the efficiency. Justify each answer mathematically using your expressions and interpret the physical reasoning behind the trend.