GATE 2024 IN Q.32: solution and answer

GATE 2024 · INQ.32 · 1 mark · NAT

A linear potentiometer (0 − 10 kΩ) is used to measure the water level as shown in the figure. The resistance between A and C varies linearly from 0 to 10 kΩ for a change in water level from 0 to 20 cm. The sensor is excited using a DC voltage source, V_S = 10 V with an internal resistance, R_S = 200 Ω. If V_out = 5 V, the water level is ______ cm (rounded off to one decimal place).

GATE 2024 IN question 32: A DC source V_S (positive terminal up) in series with a dashed-box internal resistance R_S. — Figure from the GATE 2024 IN paper. It is part of the question.

Answer and solution

ANSWER

The water level is $h = 10.2 cm$ .

Official answer: 10.1 to 10.3Source: Official answer key, GATE 2024 IN (IISc Bengaluru)

The potentiometer is linear, so the resistance from A to the wiper C is proportional to the water level $h$ : $R_{A C} = \frac{h}{20 cm} \times 10 k Ω$ .
$V_{o u t}$ is taken between the wiper C and ground (A), and nothing is connected across it, so the output is unloaded. The whole potentiometer $R_{B A} = 10 k Ω$ is therefore in series with the source resistance $R_{S}$ across $V_{S}$ .
The current is the same for every wiper position: $I = \frac{V _{S}}{R _{S} + R _{B A}} = \frac{10 V}{200 Ω + 10000 Ω} = \frac{10}{10200} A$ .
$V_{o u t}$ is the drop across the section A to C only: $V_{o u t} = I R_{A C}$ .
For $V_{o u t} = 5 V$ : $R_{A C} = \frac{5}{I} = \frac{5 \times 10200}{10} Ω = 5100 Ω = 5.1 k Ω$ .
Water level: $h = \frac{R _{A C}}{10 k Ω} \times 20 cm = \frac{5.1}{10} \times 20 cm = 10.2 cm$ .

KEY IDEA

A potentiometer with a source resistance forms a loaded divider. With an unloaded output the full track plus $R_{S}$ carry the current, so $V_{o u t} = V_{S} \frac{R _{A C}}{R _{S} + R _{p o t}}$ . Because $R_{S}$ is in series, the output cannot reach $V_{S}$ and the wiper must sit above mid-travel to give half of $V_{S}$ .

WATCH OUT FOR

Ignoring $R_{S}$ and taking $V_{o u t} = V_{S} \cdot R_{A C} / R_{p o t}$ . This gives $R_{A C} = 5 k Ω$ and $h = 10.0 cm$ , which falls outside the key range of 10.1 to 10.3. Another slip is to treat the wiper as dividing the 10 k track into two parts that both see the load. The output is unloaded, so only the total series resistance of 10.2 k sets the current.

From the GATE 2024 IN paper. Every question of this paper · Download the full paper · All 19 Sensing principles questions, with notes

GATE 2024 IN Q.32: solution and answer

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