Question

A battery of emf 21 V and internal resistance 3 $\Omega$ is connected to a resistor. If the current in the circuit is 3 A, find the resistance of the resistor.

Hint:

Remember that the internal resistance acts as a resistor strictly in series with the ideal voltage source inside the battery.

Answer 1

Concept:
• When a battery is connected to an external resistor, the total resistance of the circuit is the sum of the external resistance ($R$) and the internal resistance ($r$) of the battery.
• Ohm's law for a complete circuit relates the electromotive force (emf, $E$) to the total current ($I$) and total resistance ($R + r$).

Step 1: Identify given values
Electromotive force (emf) of the battery, $E = 21 \text{ V}$. Internal resistance of the battery, $r = 3\ \Omega$. Current flowing in the circuit, $I = 3 \text{ A}$. Let the external resistance be $R$.

Step 2: Apply Ohm's law for the complete circuit
The formula relating these quantities is: \[ I = \frac{E}{R + r} \]
Substitute the known values into the formula: \[ 3 = \frac{21}{R + 3} \]

Step 3: Solve for the external resistance $R$
Rearrange the equation to isolate $R + 3$: \[ R + 3 = \frac{21}{3} \]
\[ R + 3 = 7 \]
Subtract 3 from both sides: \[ R = 7 - 3 \]
\[ R = 4\ \Omega \]

Step 4: Conclusion
The resistance of the external resistor is $4\ \Omega$.