The resistance of a potentiometer wire of length 4 m is 10 Ω. A resistance box and 2 V accumulator are connected in series with it. What resistance should be introduced in the box to have a potential drop of \( 1\,\mu\mathrm{V/mm} \) of the potentiometer wire?
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( 4990\,\Omega \)
The potential gradient of the wire fixes the voltage across it, and so the current; the accumulator, the box and the wire are in series, so the same current flows through the box. The accumulator is taken to have negligible internal resistance.
\( \displaystyle 1\,\mu\mathrm{V/mm} = \frac{10^{-6}\,\mathrm{V}}{10^{-3}\,\mathrm{m}} = 10^{-3}\,\mathrm{V/m} \)
Voltage across the 4 m wire and the current in it:
\( V_{\mathrm{wire}} = 10^{-3}\,\mathrm{V/m} \times 4\,\mathrm{m} = 4\times10^{-3}\,\mathrm{V} \)
\( \displaystyle I = \frac{V_{\mathrm{wire}}}{R_{\mathrm{wire}}} = \frac{4\times10^{-3}\,\mathrm{V}}{10\,\Omega} = 4\times10^{-4}\,\mathrm{A} \)
The total resistance in the circuit for this current from 2 V:
\( \displaystyle R + 10\,\Omega = \frac{2\,\mathrm{V}}{4\times10^{-4}\,\mathrm{A}} = 5000\,\Omega \)
\( R = 5000 - 10 = 4990\,\Omega \)
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 – “… A current I flows through the wire which can be varied by a variable resistance (rheostat, R) in the circuit. 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 \( \phi \) is the potential drop per unit length.”