Current Electricity • NEET Revision

Example 3.42

Mixed Grouping of Cells

Question

36 cells, each of internal resistance 0.5 Ω and emf 1.5 V each, are used to send current through an external circuit of 2 Ω resistance. Find the best mode of grouping them and the current through the external circuit.

R = 2 Ω. . .. . .. . .. . .(n cells in each row)m rows

Given

Asked

Best grouping (\(m\) rows × \(n\) cells) and the maximum current \(I\).

Formula

\(I = \dfrac{mnE}{mR + nr}\)
Maximum current when \(mR = nr \;\Rightarrow\; R = \dfrac{nr}{m}\)

Solution

  1. Condition for maximum current: \(mR = nr \Rightarrow 2m = 0.5n \Rightarrow n = 4m\).
  2. With \(mn = 36\): \(m(4m) = 36 \Rightarrow m^2 = 9 \Rightarrow m = 3,\ n = 12\).
  3. Current: \(I = \dfrac{mnE}{mR + nr} = \dfrac{36 \times 1.5}{3\times 2 + 12 \times 0.5} = \dfrac{54}{12} = 4.5\ \text{A}\).
3 rows of 12 cells each; \(I = 4.5\ \text{A}\)

Short Cut

At the matched condition, \(I_{max} = \dfrac{nE}{2R} = \dfrac{12 \times 1.5}{4} = 4.5\ \text{A}\) — denominator is just \(2R\), no need to recompute the full fraction.