A cell with an electromotive force (e.m.f.) of 15.0\text{ V} and an internal resistance of… — JAMB UTME

A cell with an electromotive force (e.m.f.) of $15.0\text{ V}$ and an internal resistance of $2.0\ \Omega$ is connected across a parallel combination of two resistors with resistances of $6.0\ \Omega$ and $12.0\ \Omega$ . What is the potential difference across the parallel resistor network?

$10.0\text{ V}$ Answer
B
$15.0\text{ V}$
C
$5.0\text{ V}$
D
$13.5\text{ V}$

Answer

The potential difference across the parallel resistor network is

10.0\text{ V}

The parallel combination of

6.0\ \Omega

and

12.0\ \Omega

has an effective resistance of

4.0\ \Omega

. Adding the internal resistance of

2.0\ \Omega

gives a total circuit resistance of

6.0\ \Omega

. The circuit current is

I = \frac{15.0\text{ V}}{6.0\ \Omega} = 2.5\text{ A}

. The potential difference across the parallel load is therefore

V = 2.5\text{ A} \times 4.0\ \Omega = 10.0\text{ V}

Step-by-Step Solution

Calculate the equivalent resistance of the two parallel resistors (

R_1 = 6.0\ \Omega

and

R_2 = 12.0\ \Omega

R_p = \frac{R_1 R_2}{R_1 + R_2} = \frac{6.0 \times 12.0}{6.0 + 12.0} = \frac{72.0}{18.0} = 4.0\ \Omega

Resistors connected in parallel combine according to the reciprocal formula.

Determine the total resistance of the entire circuit including the internal resistance (

r = 2.0\ \Omega

R_{\text{total}} = R_p + r = 4.0\ \Omega + 2.0\ \Omega = 6.0\ \Omega

The cell's internal resistance is in series with the external equivalent load resistance.

Calculate the total current supplied by the cell using Ohm's law for a complete circuit.

I = \frac{E}{R_{\text{total}}} = \frac{15.0\text{ V}}{6.0\ \Omega} = 2.5\text{ A}

The current depends on the total e.m.f. divided by the total circuit resistance.

Calculate the potential difference across the parallel combination (terminal potential difference).

V = I R_p = 2.5\text{ A} \times 4.0\ \Omega = 10.0\text{ V}

The potential difference across the parallel network is the product of the total current flowing into the combination and its equivalent resistance.

Key Concept

Terminal Potential Difference and Internal Resistance

Estimated Time:1m 30s

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