Calculating Voltage Across External Resistor

This problem involves a simple electrical circuit containing a battery with a given electromotive force (EMF) and internal resistance, connected to an external resistor. We are provided with the current flowing through the circuit and need to find the voltage across the external resistor.

Understanding Circuit Concepts

• EMF ($E$): The total voltage provided by the battery when no current is drawn. Here, $E = 12$ V.
• Internal Resistance ($r$): The resistance within the battery itself. Here, $r = 3 \Omega$.
• Current ($I$): The rate of flow of charge in the circuit. Here, $I = 0.6$ A.
• External Resistor ($R_{ext}$): The resistor connected outside the battery.
• Voltage Across External Resistor ($V_{ext}$): This is also known as the terminal voltage of the battery when it's supplying current.

Applying Ohm's Law and Circuit Equations

In a circuit with a battery having internal resistance, the total voltage (EMF) is used to overcome both the external resistance and the internal resistance. Ohm's law applied to the entire circuit states:

The voltage across the external resistor ($V_{ext}$) is given by Ohm's law applied only to the external resistor:

However, we can also express the terminal voltage (voltage across the external resistor) in terms of EMF and the voltage drop across the internal resistance:

This formula is convenient because we have all the values needed ($E$, $I$, and $r$).

Step-by-Step Calculation

1. Identify Given Values:
   • EMF, $E = 12$ V
   • Internal Resistance, $r = 3 \Omega$
   • Current, $I = 0.6$ A
2. Use the Terminal Voltage Formula:

   The voltage across the external resistor is the terminal voltage ($V_{ext}$), calculated as:

   $V_{ext} = E - I \times r$
3. Substitute the Values: $V_{ext} = 12 \text{ V} - (0.6 \text{ A} \times 3 \Omega)$
4. Calculate the Voltage Drop Across Internal Resistance: $I \times r = 0.6 \text{ A} \times 3 \Omega = 1.8 \text{ V}$
5. Calculate the Final Voltage Across External Resistor: $V_{ext} = 12 \text{ V} - 1.8 \text{ V}$ $V_{ext} = 10.2 \text{ V}$

Conclusion

The voltage across the external resistor is calculated to be 10.2 V. This represents the actual potential difference available to the external circuit after accounting for the internal voltage loss within the battery.