Question

Read each question carefully. Answer all questions in the spaces provided. Use the techniques taught in this course to solve problems. Show all work. Do not skip steps. Simplify. Give exact answers unless otherwise stated. Write clearly. Make sure your steps make mathematical sense. Use correct math notation. Circle or box in answers when needed. Do not use mixed numbers. GOOD LUCK! (4) 1. (10 pts.) Find the charge on the capacitor in an RLC-series circuit when L=0.5 h,R=10Ω. C=0.01f,E(t)=150 V,a(O)=1C, and i(0)=0 A. What is the charge on the capacitor after a long time?

Expert Answer

Answer 1

After a long time, the charge on the capacitor in an RLC series circuit with specified parameters will approach its maximum value, which is calculated as the product of the circuit's capacitance and the maximum voltage, resulting in 1.5 coulombs.

Step-by-step explanation:

To find the charge on the capacitor in an RLC series circuit after a long time, we can use the fact that the charge approaches maximum charge Q = Cε, where C is the capacitance, and ε (epsilon) is the maximum voltage across the circuit. Given the circuit parameters L = 0.5 H (henry), R = 10 Ω (ohms), C = 0.01 F (farads), and E(t) = 150 V (volts), we aim to determine the charge Q on the capacitor after the system reaches steady state, i.e., after a long time.

Since the steady state scenario suggests the circuit behaves like a direct current (DC) circuit because the reactive components (inductor and capacitor) reach their maximum values, the charge on the capacitor will be Q = Cε = 0.01 F × 150 V = 1.5 C (coulombs). This is because the inductor will behave like a short circuit, and the capacitor will act like an open circuit after a very long time.

Thus, after a long time, the charge on the capacitor will reach its maximum value of 1.5 coulombs.