A battery of electromotive force (e.m.f.) 24\,\text{V} and internal resistance 2\,\Omega i… — JAMB UTME

A battery of electromotive force (e.m.f.) $24\,\text{V}$ and internal resistance $2\,\Omega$ is connected across a parallel network consisting of two resistors of resistances $6\,\Omega$ and $12\,\Omega$ . What is the total electrical energy, in Joules, dissipated in the external circuit during an operating time of $5\,\text{minutes}$ ?

Answer: 19200 J

Answer

The total electrical energy dissipated in the external circuit over 5 minutes is

19,200\,\text{J}

To compute the external energy dissipated, first combine the parallel resistors to get an equivalent external resistance of

4\,\Omega

. Adding the

2\,\Omega

internal resistance yields a total circuit resistance of

6\,\Omega

, which draws

4\,\text{A}

of current from the

24\,\text{V}

battery. The power delivered to the external load is

P_{ext} = I^2 R_p = 4^2 \times 4 = 64\,\text{W}

. Multiplying this power by the time duration in seconds (

5 \times 60 = 300\,\text{s}

) yields

19,200\,\text{J}

Step-by-Step Solution

Calculate the equivalent resistance of the external parallel circuit

R_p = \frac{R_1 \times R_2}{R_1 + R_2} = \frac{6 \times 12}{6 + 12} = \frac{72}{18} = 4\,\Omega

The two external resistors are connected in parallel.

Calculate the total current supplied by the battery

I = \frac{E}{R_p + r} = \frac{24}{4 + 2} = \frac{24}{6} = 4\,\text{A}

Ohm's law for a complete circuit accounts for both external resistance and internal resistance.

Determine the power dissipated exclusively in the external circuit

P_{ext} = I^2 R_p = (4)^2 \times 4 = 16 \times 4 = 64\,\text{W}

Electrical power dissipated across the external load depends on the total current squared times the equivalent external resistance.

Convert time from minutes to seconds and calculate total energy dissipated

t = 5 \times 60 = 300\,\text{s}

E = P_{ext} \times t = 64 \times 300 = 19,200\,\text{J}

Electrical energy is the product of power in Watts and time in seconds.

Key Concept

Electrical Energy and Power in Circuits with Internal Resistance

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