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We are given a circuit with two batteries (emf sources \( \mathcal{E}_1 = 150 \, \text{V} \) and \( \mathcal{E}_2 = 50 \, \text{V} \)), and two resistors \( R_1 = 3.0 \, \Omega \) and \( R_2 = 2.0 \, \Omega \). The potential at point \( P \) is defined as \( V_P = 130 \, \text{V} \), and we need to find the potential at point \( Q \).

### Step-by-step approach:

1. **Assume the direction of current flow**:
   - Let's assume the current flows from \( Q \) to \( P \) through the circuit.
   - Let the current through the loop be \( I \).

2. **Apply Kirchhoff's Voltage Law (KVL)**:
   Kirchhoff’s Voltage Law states that the sum of all potential differences (voltage drops and gains) in a closed loop is zero. We’ll apply this law to the loop in the circuit.

3. **Writing the KVL equation**:
   We start from point \( P \), move through \( R_2 \), and then \( \mathcal{E}_2 \), \( R_1 \), and \( \mathcal{E}_1 \), finally back to \( P \). This gives us the following KVL equation:

   \[
   V_P - I R_2 - \mathcal{E}_2 - I R_1 + \mathcal{E}_1 = V_P
   \]

   Rearranging the terms:
   
   \[
   0 = -I R_2 - \mathcal{E}_2 - I R_1 + \mathcal{E}_1
   \]

   Substitute the known values:
   
   \[
   0 = -I(2.0) - 50 - I(3.0) + 150
   \]

   Simplifying:
   
   \[
   0 = -5I + 100
   \]

   Solving for \( I \):

   \[
   I = \frac{100}{5} = 20 \, \text{A}
   \]

4. **Determine the potential at \( Q \)**:
   To find the potential at \( Q \), we calculate the voltage drop across \( R_1 \) due to the current. The potential difference between \( Q \) and \( P \) is the voltage drop across \( R_1 \).

   \[
   V_Q = V_P - I R_1 = 130 \, \text{V} - (20 \, \text{A})(3.0 \, \Omega)
   \]

   Simplifying:

   \[
   V_Q = 130 - 60 = 70 \, \text{V}
   \]

Thus, the potential at point \( Q \) is \( \boxed{70 \, \text{V}} \).

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Would you like further details or have any questions? Here are some related questions to expand the concept:

1. How would the potential at \( Q \) change if \( R_1 \) were increased?
2. What happens to the current if the battery \( \mathcal{E}_1 \) is replaced by one of a higher emf?
3. Can you calculate the power dissipated in each resistor?
4. What would be the effect of reversing the polarity of \( \mathcal{E}_2 \)?
5. How would the circuit behave if \( R_2 \) were removed?

**Tip**: In Kirchhoff's law problems, always carefully assign the signs (positive for voltage gains and negative for voltage drops) based on the assumed current direction.

In the figure, the ideal batteries have emfs 𝜀₁ = 150 V and 𝜀₂ = 50 V and the resistances are R₁ = 3.0 Ω and R₂ = 2.0 Ω. If the potential at P is defined to be 130 V, what is the potential at Q?

Learn how to apply Kirchhoff's Voltage Law (KVL) and Ohm's Law to find the potential at point Q in a circuit with two batteries and resistors. The problem involves calculating the current using given values for resistances and battery emfs.

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