An ideal battery of 6 V and resistance R are connected in series in the primary circuit of a potentiometer of length 4 m and resistance 10 Ω. The value of R, to give a potential difference of 10 mV across 20 cm of potentiometer wire is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → 290 Ω
The wire of a potentiometer carries a steady current, so the potential drop is uniform along it: the potential gradient is the same everywhere. First find the drop across the whole wire, then the current, then R from the battery loop.
Potential drop across the whole wire
\( \displaystyle \text{gradient} = \frac{10\,\mathrm{mV}}{20\,\mathrm{cm}} = 0.5\,\mathrm{mV\,cm^{-1}} \)
\( V_w = 0.5\,\mathrm{mV\,cm^{-1}}\times400\,\mathrm{cm} = 200\,\mathrm{mV} = 0.2\,\mathrm{V} \)
Current and the series resistance
\( \displaystyle I = \frac{V_w}{R_w} = \frac{0.2\,\mathrm{V}}{10\,\Omega} = 0.02\,\mathrm{A} \)
\( \displaystyle I = \frac{6\,\mathrm{V}}{R + 10\,\Omega} \Rightarrow R + 10 = \frac{6}{0.02} = 300\,\Omega \)
\( R = 290\,\Omega \)
Shortcut: The wire takes 0.2 V out of the 6 V, which is \( 1/30 \) of the battery voltage. In a series circuit the voltage divides in the ratio of the resistances, so the wire's 10 Ω is also \( 1/30 \) of the total, which is 300 Ω. Hence \( R = 300 - 10 = 290\,\Omega \).
Note: This question is from the pre-rationalised NCERT book (2022-23). The topic was removed from the latest NCERT book (2026-27), and it also appears to be out of the CUET 2026 syllabus.
NCERT (Pre-rationalised edition, 2022-23):
Potentiometer – “… Since the wire is uniform, the potential difference between A and any point at a distance l from A is
\( \displaystyle \varepsilon(l) = \phi\, l \qquad (3.89) \)
where φ is the potential drop per unit length.”