In this circuit problem, we need to find the electromotive force (emf) $\mathcal{E}$ of the ideal battery. Here’s how we can approach it:

Given:

• The current in resistor $R_6$ is $i_6 = 1.40 \, \text{A}$.
• Resistances:
  • $R_1 = R_2 = R_3 = 2.00 \, \Omega$,
  • $R_4 = 16.0 \, \Omega$,
  • $R_5 = 8.00 \, \Omega$,
  • $R_6 = 4.00 \, \Omega$.

Step 1: Analyze the circuit configuration

• Resistors $R_5$ and $R_6$ are in series because the current through them is the same. The equivalent resistance of these two resistors, $R_{56}$, is: $R_{56} = R_5 + R_6 = 8.00 \, \Omega + 4.00 \, \Omega = 12.00 \, \Omega.$

• Now, $R_4$ and $R_{56}$ are in parallel. The equivalent resistance for resistors in parallel, $R_{\text{eq}}$, is given by: $\frac{1}{R_{\text{eq}}} = \frac{1}{R_4} + \frac{1}{R_{56}} = \frac{1}{16.0 \, \Omega} + \frac{1}{12.0 \, \Omega}.$ Solving for $R_{\text{eq}}$: $R_{\text{eq}} = \frac{1}{\left( \frac{1}{16} + \frac{1}{12} \right)} = \frac{1}{\left( \frac{3}{48} + \frac{4}{48} \right)} = \frac{1}{\left( \frac{7}{48} \right)} = \frac{48}{7} \approx 6.86 \, \Omega.$

Step 2: Combine the remaining resistances

Now, resistors $R_1$, $R_2$, $R_3$, and $R_{\text{eq}}$ are all in series, so the total resistance of the circuit is: $R_{\text{total}} = R_1 + R_2 + R_3 + R_{\text{eq}} = 2.00 \, \Omega + 2.00 \, \Omega + 2.00 \, \Omega + 6.86 \, \Omega = 12.86 \, \Omega.$

Step 3: Use Ohm’s law to find the emf

The current through $R_6$ is given as $i_6 = 1.40 \, \text{A}$. Since $i_6$ flows through $R_5 + R_6$, the same current flows through the entire circuit.

Now, applying Ohm’s law: $\mathcal{E} = i_{\text{total}} \times R_{\text{total}} = 1.40 \, \text{A} \times 12.86 \, \Omega = 18.00 \, \text{V}.$

Final Answer:

The emf of the ideal battery is $\mathcal{E} = 18.00 \, \text{V}$.

Do you have any questions, or would you like more details? Here are five related questions to expand the discussion:

1. How do we calculate the total resistance for resistors in parallel and series?
2. What is the significance of using Ohm’s law in circuit analysis?
3. How would the circuit's total resistance change if we added another resistor in series with $R_1$?
4. How does the equivalent resistance change when two resistors are in parallel?
5. How does the current distribution behave in parallel and series configurations?

Tip: When solving for equivalent resistance in parallel, always remember the formula involves reciprocal sums, which often results in a lower equivalent resistance than any individual resistor in the combination.