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To solve this problem using Kirchhoff's Voltage Law (KVL), we need to apply the loop rule, which states that the sum of all potential differences around a closed loop is equal to zero.

### Analysis of the given circuit
- The circuit includes:
  - An 18V battery \( \mathcal{E}_1 = 18V \)
  - A second battery with an unknown voltage \( \mathcal{E}_2 \)
  - Three resistors \( R_1 = 0.5 \, \Omega \), \( R_2 = 2.5 \, \Omega \), and \( R_3 = 1.5 \, \Omega \)
- The currents in the circuit are labeled \( I_1 \), \( I_2 \), and \( I_3 \), and their directions are indicated.

### Loop Equation for Loop abcdefgha
Let's traverse the loop in the clockwise direction from point \( a \) through \( b \), \( c \), \( d \), \( e \), \( f \), \( g \), \( h \), and back to \( a \). We'll use the following conventions:
- Voltage drops across resistors in the direction of current flow are \( -IR \).
- Voltage rises across batteries are taken as positive when going from negative to positive terminal.

**Loop: abcdefgha**
1. From \( a \) to \( b \), there is a voltage rise across the 18V battery: \( +18V \).
2. From \( b \) to \( c \), there is a voltage drop across \( R_2 \): \( -I_2 R_2 \).
3. From \( c \) to \( d \), a voltage drop across \( R_1 \): \( -I_1 R_1 \).
4. From \( d \) to \( e \), no component, so no voltage change.
5. From \( e \) to \( f \), there is a voltage drop across \( R_3 \): \( -I_3 R_3 \).
6. From \( f \) to \( g \), a voltage rise across the second battery \( \mathcal{E}_2 \): \( +\mathcal{E}_2 \).
7. From \( g \) to \( h \), there is a voltage drop across \( R_1 \): \( -I_1 R_1 \).
8. Returning from \( h \) to \( a \), no component is encountered.

Adding these contributions and setting the total to zero:
\[
18V - I_2 R_2 - I_1 R_1 - I_3 R_3 + \mathcal{E}_2 - I_1 R_1 = 0.
\]

This equation can be simplified to:
\[
18V + \mathcal{E}_2 - I_2 (2.5 \, \Omega) - I_1 (2 \cdot 0.5 \, \Omega) - I_3 (1.5 \, \Omega) = 0.
\]

Would you like further explanation on any step or need help with related questions?

### Related Questions:
1. How do we apply Kirchhoff's Current Law to this circuit?
2. What happens to the loop equation if the current directions are reversed?
3. How would adding another resistor in series affect the loop equation?
4. Can the loop rule be applied to circuits with only capacitors?
5. How can Kirchhoff's rules be used to find power dissipated in each resistor?

**Tip:** Always assign current directions consistently when applying Kirchhoff's Voltage Law. If you guess wrong, the current value will just be negative.

Consider the circuit diagram and apply the loop rule to the loop abcdefgha in terms of the variables in the figure.

Learn how to apply Kirchhoff's Voltage Law (KVL) to a complex circuit with two batteries and three resistors. This detailed step-by-step solution explains how to formulate the loop equation, taking into account current directions and potential differences across components. Suitable for college-level physics or electrical engineering students.

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