The problem asks to find the resistance of $R_4$ in a circuit where the meter reads 7.84 mA (milliamperes) when the switch is in position A, and the voltage source is 12 V.

Step-by-step approach:

1. Circuit Configuration:

   • Resistors $R_1 = 220 \ \Omega$, $R_2 = 470 \ \Omega$, and $R_3 = 510 \ \Omega$ are arranged as shown in the diagram.
   • The ammeter reads 7.84 mA of current.

2. Total Resistance in the Circuit: Since the circuit is powered by a 12 V source and the current through the circuit is 7.84 mA, we can use Ohm's law to calculate the total resistance of the circuit: $I = \frac{V}{R_{\text{total}}} \quad \Rightarrow \quad R_{\text{total}} = \frac{V}{I}$ Here, $V = 12 \ \text{V}, \quad I = 7.84 \ \text{mA} = 7.84 \times 10^{-3} \ \text{A}$ Substituting the values: $R_{\text{total}} = \frac{12 \ \text{V}}{7.84 \times 10^{-3} \ \text{A}} = 1531.63 \ \Omega$

3. Resistors in Series and Parallel:

   • The circuit has $R_1$, $R_2$, and $R_3$ in some combination (possibly a mix of series and parallel).
   • Let’s consider how $R_4$ fits in: The total resistance $R_{\text{total}}$ is a combination of $R_1$, $R_2$, $R_3$, and $R_4$.

4. Finding $R_4$: We now need to use the known resistances of $R_1$, $R_2$, and $R_3$ and subtract them from the total resistance to find $R_4$.

Would you like a more detailed breakdown of how to handle the resistors in series/parallel and derive $R_4$? Let me know if you want to continue or if you have any questions!

Related Questions:

1. How does the position of the switch affect the circuit configuration?
2. How can you calculate the equivalent resistance of resistors in parallel?
3. Why is the current given in milliamperes important for this calculation?
4. What would happen to the total resistance if $R_4$ were removed from the circuit?
5. How does the total resistance change if the power source voltage increases?

Tip:

When dealing with circuits that combine resistors in series and parallel, always simplify one part of the circuit at a time by calculating the equivalent resistance step by step.