Case: Read the passage and answer the question.
A cell of unknown emf E and internal resistance r, two unknown resistances \( R_1 \) and \( R_2 \) \( (R_2\gt R_1) \) and a perfect ammeter are given. The current in the circuit is measured in five different situations:
(i) without any external resistance in the circuit,
(ii) with resistance \( R_1 \) only,
(iii) with resistance \( R_2 \) only,
(iv) with both \( R_1 \) and \( R_2 \) used in series combination, and
(v) with both \( R_1 \) and \( R_2 \) used in parallel combination.
The currents obtained in the five cases are 0.42 A, 0.6 A, 1.05 A, 1.4 A and 4.2 A, but not necessarily in that order.
Consider the following statements:
(A) Two cells like this one, joined in series, have an equivalent internal resistance of \( r/2 \).
(B) The internal resistance has the same value for all cells.
(C) The internal resistance of a cell may be neglected in calculations when \( E \) is much larger than \( Ir \).
(D) Two cells like this one, joined in series with the positive terminal of one to the negative terminal of the other, have an equivalent emf of \( 2E \).
Which of the above statements are incorrect? Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
For cells in series, the emfs add when the current leaves each cell from its positive electrode. The internal resistances always add.
(A) Two cells like this one, joined in series, have an equivalent internal resistance of \( r/2 \). — False. In series the internal resistances add, so for two equal cells \( r_{eq}=2r \). A value smaller than \( r \) is found only for cells in parallel.
(B) The internal resistance has the same value for all cells. — False. The internal resistance is different for different cells. For example, a dry cell has a much higher internal resistance than a common electrolytic cell.
(C) The internal resistance of a cell may be neglected in calculations when \( E \) is much larger than \( Ir \). — True. When \( Ir \) is very small compared with \( E \), the terminal potential difference \( E-Ir \) is almost equal to \( E \).
(D) Two cells like this one, joined in series with the positive terminal of one to the negative terminal of the other, have an equivalent emf of \( 2E \). — True. With this connection the current leaves each cell from its positive electrode, so the two emfs add to give \( E_{eq}=2E \).
NCERT (2026-27):
Cells in Series and in Parallel – “… Comparing the last two equations, we get εeq = ε1 + ε2 … and req = r1 + r2”
Cells in Series and in Parallel – “… This is so, when the current leaves each cell from the positive electrode. …”
Cells, Emf, Internal Resistance – “In practical calculations, internal resistances of cells in the circuit may be neglected when the current I is such that ε >> I r. The actual values of the internal resistances of cells vary from cell to cell. The internal resistance of dry cells, however, is much higher than the common electrolytic cells.”