This is a circuit analysis problem. We are asked to find the potential difference (voltage drop) across the 2-ohm resistor ($R_3 = 2 \, \Omega$) in the circuit. Here's how we can proceed step-by-step:

Step 1: Identify the configuration of the circuit.

• There are two voltage sources: $\varepsilon_1 = 12 \, V$ and $\varepsilon_2 = 18 \, V$.
• The resistors $R_1 = 6 \, \Omega$, $R_2 = 4 \, \Omega$, and $R_3 = 2 \, \Omega$ are connected in a combination of series.

Step 2: Combine the resistors.

The three resistors are in series, so the total resistance $R_{\text{total}}$ is the sum of all resistances: $R_{\text{total}} = R_1 + R_2 + R_3 = 6 \, \Omega + 4 \, \Omega + 2 \, \Omega = 12 \, \Omega$

Step 3: Calculate the total voltage.

The two voltage sources are in series, so the total voltage $V_{\text{total}}$ is the sum of the two voltages: $V_{\text{total}} = 12 \, V + 18 \, V = 30 \, V$

Step 4: Find the total current in the circuit.

Using Ohm's law, $I = \frac{V}{R}$, the total current $I_{\text{total}}$ flowing through the circuit is: $I_{\text{total}} = \frac{V_{\text{total}}}{R_{\text{total}}} = \frac{30 \, V}{12 \, \Omega} = 2.5 \, \text{A}$

Step 5: Calculate the voltage drop across the 2-ohm resistor.

The voltage drop across $R_3$ can be calculated using Ohm's law, $V = IR$, where $I$ is the current and $R$ is the resistance. Thus, the voltage across $R_3$ is: $V_{R_3} = I_{\text{total}} \times R_3 = 2.5 \, \text{A} \times 2 \, \Omega = 5 \, V$

Final Answer:

The potential difference (voltage drop) across the 2-ohm resistor is 5 V.

Would you like further details on this calculation or any clarification?

Related Questions:

1. How do you calculate the total resistance in a parallel circuit?
2. What is the difference between series and parallel circuits?
3. How do you apply Kirchhoff's Voltage Law to this type of circuit?
4. How do you calculate the power dissipated by the 2-ohm resistor?
5. What happens if one of the voltage sources is removed in this circuit?

Tip:

When dealing with circuits, always check whether the components are connected in series or parallel, as this greatly affects how you sum resistances and calculate the total current.