A battery with an electromotive force (e.m.f.) of 12.0\text{ V} and an internal resistance… — JAMB UTME

A battery with an electromotive force (e.m.f.) of $12.0\text{ V}$ and an internal resistance of $1.0\text{ }\Omega$ is connected to a circuit containing two resistors, $6.0\text{ }\Omega$ and $3.0\text{ }\Omega$ , connected in parallel with each other. What is the potential difference across the $3.0\text{ }\Omega$ resistor?

$8.0\text{ V}$ Answer
B
$12.0\text{ V}$
C
$4.0\text{ V}$
D
$3.6\text{ V}$

Answer

The potential difference across the

3.0\text{ }\Omega

resistor is

8.0\text{ V}

The parallel external resistors of

6.0\text{ }\Omega

and

3.0\text{ }\Omega

combine to give an equivalent resistance of

2.0\text{ }\Omega

. Including the internal resistance of

1.0\text{ }\Omega

, the total circuit resistance is

3.0\text{ }\Omega

. This leads to a total circuit current of

4.0\text{ A}

. Multiplying this current by the external equivalent resistance of

2.0\text{ }\Omega

gives the terminal voltage of

8.0\text{ V}

, which is identical across both parallel branches.

Step-by-Step Solution

Calculate the equivalent resistance (

R_p

) of the parallel combination.

R_p = \frac{6.0 \times 3.0}{6.0 + 3.0} = \frac{18.0}{9.0} = 2.0\text{ }\Omega

Resistors in parallel combine according to reciprocal addition.

Find the total circuit resistance (

R_{\text{total}}

) including the internal resistance (

r

R_{\text{total}} = R_p + r = 2.0\text{ }\Omega + 1.0\text{ }\Omega = 3.0\text{ }\Omega

The internal resistance of the power supply acts in series with the external equivalent load resistance.

Calculate the total current (

I

) drawn from the battery using Ohm's law for a complete circuit.

I = \frac{E}{R_{\text{total}}} = \frac{12.0\text{ V}}{3.0\text{ }\Omega} = 4.0\text{ A}

Current is determined by the total electromotive force divided by the total resistance of the circuit.

Determine the terminal potential difference (

V

) across the parallel network.

V = I \times R_p = 4.0\text{ A} \times 2.0\text{ }\Omega = 8.0\text{ V}

Components connected in parallel experience the same potential difference, which equals the terminal voltage of the battery.

Key Concept

Terminal potential difference in circuits with internal resistance

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