Let's solve the circuit step-by-step: Given: - \( R_1 = 4.00\,\Omega \) - \( R_2 = 8.00\,\Omega \) - \( V_s = 24.0\angle 60^\circ \) V - \( L = j6.00\,\Omega \) - \( C = -j4.00\,\Omega \) ### Step 1: **Find Total Impedance** The circuit is a series circuit, so the total impedance is: \[ Z_{total} = R_1 + j6 + R_2 - j4 \] \[ = (4 + 8) + (j6 - j4) \] \[ = 12 + j2~\Omega \] ### Step 2: **Find Total Current \( I_1 \)** Using Ohm's Law: \[ I_1 = \frac{V_s}{Z_{total}} \] First, find the magnitude and angle of \( Z_{total} \): \[ |Z_{total}| = \sqrt{12^2 + 2^2} = \sqrt{144 + 4} = \sqrt{148} \approx 12.17~\Omega \] \[ \theta_Z = \arctan\left(\frac{2}{12}\right) = \arctan\left(\frac{1}{6}\right) \approx 9.46^\circ \] So, \[ Z_{total} = 12.17\angle 9.46^\circ~\Omega \] \[ I_1 = \frac{24\angle 60^\circ}{12.17\angle 9.46^\circ} = 1.973\angle(60^\circ - 9.46^\circ) \] \[ I_1 = 1.973\angle 50.54^\circ~\text{A} \] ### Step 3: **Find Voltages and Currents** #### (a) **Voltage across \( R_1 + L \) (i.e., \( V_1 \))** The impedance here is \( Z_1 = R_1 + j6 = 4 + j6 = 7.21\angle 56.31^\circ~\Omega \). \[ V_1 = I_1 \cdot Z_1 = 1.973\angle 50.54^\circ \times 7.21\angle 56.31^\circ \] \[ = (1.973 \times 7.21) \angle (50.54^\circ + 56.31^\circ) \] \[ = 14.23 \angle 106.85^\circ~\text{V} \] #### (b) **Voltage across \( R_2 + C \) (i.e., \( V_2 \))** The impedance here is \( Z_2 = R_2 - j4 = 8 - j4 = 8.94\angle -26.57^\circ~\Omega \). \[ V_2 = I_1 \cdot Z_2 = 1.973\angle 50.54^\circ \times 8.94\angle -26.57^\circ \] \[ = (1.973 \times 8.94) \angle (50.54^\circ - 26.57^\circ) \] \[ = 17.64\angle 23.97^\circ~\text{V} \] #### (c) **Current through Each Element** Since it's a series circuit: \[ I_1 = I_2 = I_3 = 1.973\angle 50.54^\circ~\text{A} \] --- ### **Final Answers** #### **Currents:** - \( I_1 = I_2 = I_3 = \boxed{1.97\angle 50.5^\circ~\text{A}} \) #### **Voltages:** - \( V_1 = \boxed{14.23\angle 106.9^\circ~\text{V}} \) (across \( R_1 + L \)) - \( V_2 = \boxed{17.64\angle 24.0^\circ~\text{V}} \) (across \( R_2 + C \)) --- **Let me know if you need a phasor diagram or further breakdown!**