Let's solve the problem **step by step**. --- ### **Step 1: Identify Given Values** - **EMF equation:** \(\mathcal{E} = 40V e^{i(15 \, \text{rad/sec})t}\) - **Capacitance, \(C\):** \(10 \text{ mF} = 10 \times 10^{-3} \text{ F}\) - **Inductance, \(L\):** \(100 \text{ mH} = 0.1 \text{ H}\) - **Resistors:** \(3\,\Omega\) and \(2\,\Omega\) (in series, total resistance = \(3\,\Omega + 2\,\Omega = 5\,\Omega\)) - **Angular frequency:** \(\omega = 15 \, \text{rad/sec}\) - **Peak voltage:** \(V_0 = 40\,V\) --- ### **Step 2: Find Total Impedance \(Z\) of the Circuit** The circuit is a **series RLC circuit**. - **Total Resistance:** \(R = 3 + 2 = 5\,\Omega\) - **Inductive Reactance:** \(X_L = \omega L = 15 \times 0.1 = 1.5\,\Omega\) - **Capacitive Reactance:** \(X_C = \frac{1}{\omega C} = \frac{1}{15 \times 0.01} = \frac{1}{0.15} \approx 6.67\,\Omega\) - **Net Reactance:** \(X = X_L - X_C = 1.5 - 6.67 = -5.17\,\Omega\) - **Impedance:** \[ Z = \sqrt{R^2 + X^2} = \sqrt{5^2 + (-5.17)^2} = \sqrt{25 + 26.73} = \sqrt{51.73} \approx 7.19\,\Omega \] --- ### **Step 3: Find the RMS Value of the EMF** - **Peak Value, \(V_0 = 40\,V\)** - **RMS Value, \(V_{rms} = \frac{V_0}{\sqrt{2}} = \frac{40}{\sqrt{2}} = 28.28\,V\)** --- ### **Step 4: Calculate Average Power Supplied** The average power in an AC circuit is: \[ P_{avg} = \frac{V_{rms}^2}{Z} \cos\phi \] where \(\cos\phi\) is the power factor: \[ \cos\phi = \frac{R}{Z} \] So, \[ P_{avg} = \frac{V_{rms}^2}{Z} \cdot \frac{R}{Z} = V_{rms}^2 \cdot \frac{R}{Z^2} \] Let's plug in the numbers: - \(V_{rms} = 28.28\,V\) - \(R = 5\,\Omega\) - \(Z \approx 7.19\,\Omega\) \[ P_{avg} = (28.28)^2 \cdot \frac{5}{(7.19)^2} \] Calculate: - \((28.28)^2 = 800\) - \((7.19)^2 \approx 51.7\) - \(\frac{5}{51.7} \approx 0.0967\) \[ P_{avg} = 800 \times 0.0967 \approx 77.4\,W \] --- ## **Final Answer** \[ \boxed{77.4\,\text{W}} \] This is the **average power** supplied by the EMF to the electric circuit.