https://file.aiquickdraw.com/mathbot/file/1728134480523txz9r0kn.jpg

To find the electromotive force (emf) of the ideal battery in this circuit, we can use Ohm's Law and Kirchhoff's Loop Rule. Let's break it down step by step:

### Circuit Configuration:
- The circuit has 6 resistors: \(R_1\), \(R_2\), \(R_3\), \(R_4\), \(R_5\), and \(R_6\).
- The current \(i_6 = 1.31 \, \text{A}\) flows through \(R_6 = 4.65 \, \Omega\).

### Given Values:
- \(R_1 = R_2 = R_3 = 1.68 \, \Omega\)
- \(R_4 = 17.7 \, \Omega\)
- \(R_5 = 7.71 \, \Omega\)
- \(R_6 = 4.65 \, \Omega\)
- \(i_6 = 1.31 \, \text{A}\)

### Step 1: Calculate Voltage Drop across \(R_6\)
The voltage across \(R_6\) is given by Ohm's Law:
\[
V_6 = i_6 \cdot R_6 = 1.31 \, \text{A} \times 4.65 \, \Omega = 6.0915 \, \text{V}
\]

### Step 2: Simplify the Circuit
- \(R_1\), \(R_2\), and \(R_3\) are in series, so their equivalent resistance is:
\[
R_{123} = R_1 + R_2 + R_3 = 1.68 \, \Omega + 1.68 \, \Omega + 1.68 \, \Omega = 5.04 \, \Omega
\]
- \(R_4\), \(R_5\), and \(R_6\) are in series, so their equivalent resistance is:
\[
R_{456} = R_4 + R_5 + R_6 = 17.7 \, \Omega + 7.71 \, \Omega + 4.65 \, \Omega = 30.06 \, \Omega
\]

### Step 3: Total Resistance of the Circuit
Since the two series combinations are in parallel, the total resistance \(R_{\text{total}}\) is:
\[
\frac{1}{R_{\text{total}}} = \frac{1}{R_{123}} + \frac{1}{R_{456}} = \frac{1}{5.04} + \frac{1}{30.06}
\]
Solving this:
\[
\frac{1}{R_{\text{total}}} \approx 0.1984 + 0.0333 = 0.2317
\]
\[
R_{\text{total}} \approx 4.32 \, \Omega
\]

### Step 4: Apply Ohm's Law to Find EMF
Using Ohm's Law for the whole circuit, we can calculate the emf (\(\mathcal{E}\)) by multiplying the total resistance by the current \(i_6\), which is the same through the entire circuit:
\[
\mathcal{E} = i_6 \cdot R_{\text{total}} = 1.31 \, \text{A} \times 4.32 \, \Omega = 5.66 \, \text{V}
\]

Thus, the emf of the ideal battery is approximately **5.66 V**.

Would you like further clarification or details?

### Related Questions:
1. How does changing \(R_6\) affect the total resistance of the circuit?
2. What would happen if \(R_4\) were removed from the circuit?
3. How would the total current change if \(R_5\) were doubled?
4. What is the significance of the parallel configuration in determining the total resistance?
5. Can you calculate the power dissipated in \(R_6\)?

### Tip:
When analyzing complex circuits, simplifying series and parallel resistances step by step can help make the analysis much easier!

In the figure, the current in resistance R6 is i6 = 1.31 A and the resistances are R1 = R2 = R3 = 1.68Ω, R4 = 17.7Ω, R5 = 7.71Ω, and R6 = 4.65Ω. What is the emf of the ideal battery?

Learn how to calculate the electromotive force (EMF) of a battery in a circuit with resistances R1 = R2 = R3 = 1.68Ω, R4 = 17.7Ω, R5 = 7.71Ω, and R6 = 4.65Ω. The current through R6 is i6 = 1.31 A. This problem involves Ohm's Law and Kirchhoff's Loop Rule.

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