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This problem involves an **RC circuit** where resistors \( R_1 \), \( R_2 \), and \( R_3 \) are arranged in a combination, and we are asked to determine the **time constant** for the circuit. The given values are:

- \( R_1 = 6.0 \, \Omega \)
- \( R_2 = 15 \, \Omega \)
- \( R_3 = 30 \, \Omega \)
- \( C = 79 \, \mu F = 79 \times 10^{-6} \, F \)
- \( \mathcal{E} = 12 \, V \) (though the voltage is not needed for time constant calculations).

### Step 1: Determine the Equivalent Resistance

The time constant \( \tau \) of an RC circuit is calculated using the formula:
\[
\tau = R_{\text{eq}} C
\]
where \( R_{\text{eq}} \) is the equivalent resistance seen by the capacitor, and \( C \) is the capacitance.

From the diagram, we see that \( R_1 \) and \( R_2 \) are in parallel, and this combination is in series with \( R_3 \). The equivalent resistance is found by:

#### 1.1: Calculate the Parallel Combination of \( R_1 \) and \( R_2 \):
\[
R_{12} = \frac{R_1 R_2}{R_1 + R_2} = \frac{(6.0)(15)}{6.0 + 15} = \frac{90}{21} \approx 4.29 \, \Omega
\]

#### 1.2: Add the Series Resistor \( R_3 \):
\[
R_{\text{eq}} = R_{12} + R_3 = 4.29 + 30 = 34.29 \, \Omega
\]

### Step 2: Calculate the Time Constant \( \tau \)
\[
\tau = R_{\text{eq}} C = (34.29 \, \Omega)(79 \times 10^{-6} \, F) = 0.00271 \, \text{seconds} = 2.71 \, \text{ms}
\]

Thus, the time constant \( \tau \) is **2.71 milliseconds**.

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Would you like more details or have any questions about this solution? 

Here are some related questions for further understanding:
1. How do parallel and series resistors affect the total resistance in a circuit?
2. What happens to the time constant if the capacitance is doubled?
3. Why is the voltage \( \mathcal{E} \) not needed to find the time constant in this RC circuit?
4. How does the time constant affect the charging and discharging rate of the capacitor?
5. What are the practical applications of RC circuits in electronics?

**Tip:** The time constant \( \tau \) determines how quickly a capacitor charges or discharges. After a time of \( 5\tau \), a capacitor is considered fully charged or discharged.

Find the time constant for the RC circuit with R1 = 6.0Ω, R2 = 15Ω, R3 = 30Ω, C = 79μF, and E = 12V.

Learn how to calculate the time constant for an RC circuit with resistors arranged in both parallel and series configurations. Given values include R1 = 6.0Ω, R2 = 15Ω, R3 = 30Ω, and capacitance C = 79μF. A detailed step-by-step solution shows how to find the equivalent resistance and time constant, suitable for university-level physics or engineering students.

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