A potentiometer wire of length 100\text{ cm} has a resistance of 5.0\ \Omega. It is connec… — JAMB UTME

A potentiometer wire of length $100\text{ cm}$ has a resistance of $5.0\ \Omega$ . It is connected in series with a driver cell of e.m.f. $3.0\text{ V}$ (having negligible internal resistance) and a protective series resistor $R_s$ . A balance point is obtained at $60.0\text{ cm}$ along the wire for a cell of e.m.f. $1.2\text{ V}$ . What is the value of the series resistor $R_s$ in ohms?

Answer: 2.5 \Omega

Answer

The resistance of the series resistor

R_s

2.5\ \Omega

At balance, the potential drop across the

60.0\text{ cm}

portion of the potentiometer wire equals the test cell e.m.f. (

1.2\text{ V}

). Since the

100\text{ cm}

wire has a total resistance of

5.0\ \Omega

, the

60.0\text{ cm}

section has a resistance of

3.0\ \Omega

. This requires a main circuit current of

I = \frac{1.2\text{ V}}{3.0\ \Omega} = 0.4\text{ A}

. The total driver circuit resistance is

R_{total} = \frac{3.0\text{ V}}{0.4\text{ A}} = 7.5\ \Omega

. Subtracting the wire's

5.0\ \Omega

resistance gives

R_s = 2.5\ \Omega

Step-by-Step Solution

Calculate the resistance of the balanced portion of the potentiometer wire.

R_{balance} = 5.0\ \Omega \times \frac{60.0\text{ cm}}{100\text{ cm}} = 3.0\ \Omega

The resistance of a uniform wire is directly proportional to its length.

Determine the current in the main potentiometer circuit.

I = \frac{1.2\text{ V}}{3.0\ \Omega} = 0.4\text{ A}

At the balance point, no current flows through the galvanometer branch, so the potential difference across the balance length equals the e.m.f. of the test cell.

Compute the total resistance of the driver circuit.

R_{total} = \frac{3.0\text{ V}}{0.4\text{ A}} = 7.5\ \Omega

According to Ohm's law, total resistance equals the total e.m.f. divided by the circuit current.

Calculate the value of the series resistor

R_s

R_s = 7.5\ \Omega - 5.0\ \Omega = 2.5\ \Omega

The total resistance is the sum of the potentiometer wire resistance and the series resistance.

Key Concept

Potentiometer balance condition and circuit analysis

Estimated Time:1m 30s

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