Consider N = n₁ n₂ identical cells, each having an emf E and an internal resistance r . Assume n₁ cells are connected in series to form a line, and n₂ such lines are connected in parallel. This combination supplies current to an external resistance R . Assuming n₁ and n₂ can be varied continuously, determine the relation between n₁ , n₂ , R , and r that yields the maximum current in R .

Step-by-step solution

Equivalent emf of one row of n₁ cells in series is n₁ E and its internal resistance is n₁ r . Since n₂ such rows are connected in parallel, the equivalent emf of the entire combination remains E_ eq = n₁ E and the equivalent internal resistance is r_ eq = n₁ r n₂ . The current I through the external resistance R is given by: I = E_ eq R + r_ eq = n₁ E R + n₁ r n₂ = n₁ n₂ E n₂ R + n₁ r Since the total number of cells N = n₁ n₂ is constant, the numerator N E is constant. To maximize the current I , the denominator n₂