A DC power source with an electromotive force (e.m.f.) of E = 12.0\text{ V} and an interna… — JAMB UTME

A DC power source with an electromotive force (e.m.f.) of $E = 12.0\text{ V}$ and an internal resistance of $r = 1.0\text{ }\Omega$ is connected to an external load. The load consists of two parallel resistors, $R_1 = 3.0\text{ }\Omega$ and $R_2 = 6.0\text{ }\Omega$ , connected in series with an unknown resistor $R_x$ . A real voltmeter with an internal resistance of $R_v = 90.0\text{ }\Omega$ is placed directly across the terminals of the power source and reads $V = 10.8\text{ V}$ . Calculate the resistance of $R_x$ in ohms.

Answer: 8 \Omega

Answer

The resistance of

R_x

8.0\text{ }\Omega

Applying the relationship

V = E - Ir

yields a total current of

1.2\text{ A}

through the cell. The total external resistance connected across the terminals is

9.0\text{ }\Omega

. Subtracting the parallel admittance of the

90.0\text{ }\Omega

voltmeter yields a load resistance of

10.0\text{ }\Omega

. Subtracting the

2.0\text{ }\Omega

equivalent resistance of the parallel pair (

3.0\text{ }\Omega

and

6.0\text{ }\Omega

) gives

8.0\text{ }\Omega

for

R_x

Step-by-Step Solution

Calculate total current supplied by the cell

I = 1.2\text{ A}

Terminal voltage is related to battery e.m.f. and internal resistance by

V = E - Ir

Calculate equivalent resistance of the entire external circuit across terminals

R_{\text{ext}} = 9.0\text{ }\Omega

By Ohm's law,

R_{\text{ext}} = \frac{V}{I} = \frac{10.8\text{ V}}{1.2\text{ A}} = 9.0\text{ }\Omega

Determine the resistance of the main circuit load

R_L

R_L = 10.0\text{ }\Omega

The voltmeter is in parallel with

R_L

, giving

\frac{1}{R_{\text{ext}}} = \frac{1}{R_v} + \frac{1}{R_L}

Calculate the equivalent resistance

R_p

of the parallel combination of

R_1

and

R_2

R_p = 2.0\text{ }\Omega

R_p = \frac{R_1 R_2}{R_1 + R_2} = \frac{3.0 \times 6.0}{3.0 + 6.0} = 2.0\text{ }\Omega

Determine the value of

R_x

R_x = 8.0\text{ }\Omega

The main circuit load consists of

R_p

in series with

R_x

, so

R_L = R_p + R_x

Key Concept

Terminal potential difference, loading effect of measuring instruments, and resistor network analysis

Estimated Time:2m 30s

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