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The problem asks us to find:

- **(a)** the current through resistor \( R_1 \),
- **(b)** the current through resistor \( R_2 \),
- **(c)** the potential difference between points \( a \) and \( b \), 

given that:
- \( R_1 = 140 \, \Omega \),
- \( R_2 = 69 \, \Omega \),
- \( \mathcal{E}_1 = 7.0 \, \text{V} \) (voltage of first battery),
- \( \mathcal{E}_2 = 4.0 \, \text{V} \) (voltage of second battery),
- \( \mathcal{E}_3 = 5.0 \, \text{V} \) (voltage of third battery).

### Applying Kirchhoff's Law:

We'll approach this problem by using Kirchhoff's loop and junction rules. 

1. **Kirchhoff's Junction Rule**: The total current entering a junction equals the total current leaving the junction.
2. **Kirchhoff's Loop Rule**: The sum of potential differences in a closed loop is zero.

Let's define the currents:
- Let \( I_1 \) be the current through \( R_1 \) (for part (a)),
- Let \( I_2 \) be the current through \( R_2 \) (for part (b)).

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I see you've entered the values for the currents:

- \( I_1 \approx 0.02857 \, \text{A} \),
- \( I_2 \approx 0.02899 \, \text{A} \).

Now, the final task (part (c)) is to determine the **potential difference** between points \( a \) and \( b \). I'll calculate it assuming this is a straightforward application of Kirchhoff's laws.

Would you like me to proceed with the detailed steps for solving (c), or would you like a breakdown of how to reach the answers for parts (a) and (b) too? Let me know if you want a complete solution for (c)! 

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Here are some relative questions:

1. How is Kirchhoff's law useful in analyzing complex circuits like this?
2. Why do we need two laws (junction and loop rules) to solve for currents in circuits?
3. How do different battery configurations affect potential differences in a circuit?
4. What is the significance of resistances in determining the current in each branch?
5. How would you calculate the power dissipated by each resistor in this circuit?

**Tip**: Kirchhoff's laws are powerful tools for solving any circuit. Start by applying the junction rule to split currents at any node, then use the loop rule to find the relations between currents and potential differences in each loop.

In the figure, R1 = 140 Ω, R2 = 69 Ω, and the ideal batteries have emfs E1 = 7.0 V, E2 = 4.0 V, and E3 = 5.0 V. Find (a) the current in R1, (b) the current in R2, and (c) the potential difference between points a and b.

Solve for the currents in R1 and R2, and the potential difference between points a and b using Kirchhoff's Laws in a circuit with emfs of 7.0 V, 4.0 V, and 5.0 V and resistances of 140 Ω and 69 Ω.

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