The circuit provided appears to be a combination of series and parallel resistances. We have the following key information from the circuit:

• $R_1 = 10 \, \Omega$
• $R_2 = 20 \, \Omega$
• $R_3 = 20 \, \Omega$
• $R_4$ is unknown.
• $V_3 = 20 \, V$ across $R_3$
• The total voltage $V_T = 120 \, V$

Steps to calculate $R_4$:

1. Calculate the current through $R_3$: Using Ohm's law, the current through $R_3$ is given by: $I_3 = \frac{V_3}{R_3} = \frac{20 \, V}{20 \, \Omega} = 1 \, A$ Since $R_2$ and $R_3$ are in parallel, the current through $R_2$ will also be 1 A (because they have the same resistance).

2. Total current through the parallel combination of $R_2$ and $R_3$: $I_{\text{parallel}} = I_2 + I_3 = 1 \, A + 1 \, A = 2 \, A$

3. Voltage across the parallel combination: Since the total current through the parallel combination is 2 A and the equivalent resistance of the parallel branch is: $R_{\text{parallel}} = \frac{R_2 R_3}{R_2 + R_3} = \frac{20 \times 20}{20 + 20} = 10 \, \Omega$ The voltage across the parallel combination is: $V_{\text{parallel}} = I_{\text{parallel}} \times R_{\text{parallel}} = 2 \, A \times 10 \, \Omega = 20 \, V$

4. Voltage across $R_1$: The total voltage $V_T = 120 \, V$, and the voltage across the parallel branch is 20 V. Therefore, the voltage across $R_1$ is: $V_1 = V_T - V_{\text{parallel}} = 120 \, V - 20 \, V = 100 \, V$

5. Current through $R_1$: Using Ohm's law, the current through $R_1$ is: $I_1 = \frac{V_1}{R_1} = \frac{100 \, V}{10 \, \Omega} = 10 \, A$

6. Current through $R_4$: The current entering the junction before $R_2$, $R_3$, and $R_4$ is 10 A. The current through the parallel branch is 2 A, so the current through $R_4$ is: $I_4 = I_1 - I_{\text{parallel}} = 10 \, A - 2 \, A = 8 \, A$

7. Calculate $R_4$: Using Ohm's law for $R_4$, where the voltage across $R_4$ is $V_{\text{parallel}} = 20 \, V$: $R_4 = \frac{V_{\text{parallel}}}{I_4} = \frac{20 \, V}{8 \, A} = 2.5 \, \Omega$

Thus, the value of $R_4$ is 2.5 Ω.

Would you like further details or clarification?

Here are some related questions:

1. How would the circuit change if $R_4$ were increased?
2. How can we use Kirchhoff’s laws to solve this circuit?
3. What happens to the total current if we reduce $R_2$ and $R_3$?
4. What is the impact of adding another resistor in series with $R_1$?
5. How would you approach this problem using Thevenin’s theorem?

Tip: When dealing with parallel resistors, always calculate the equivalent resistance first to simplify the analysis.