Final answer:
To find v, UR, UL, and UC at t = 20.0 ms, calculate the impedance and the current in the circuit. Then, calculate the voltage drops across each component using Ohm's law. Finally, compare the sum of the voltage drops to the source voltage.
Step-by-step explanation:
To find v, UR, UL, and UC at t = 20.0 ms, we need to calculate the impedance and the current in the circuit. The impedance (Z) can be calculated as the square root of the sum of the squared resistor (R) and the squared reactance (X), where X = XL - XC. Here, XL is the inductive reactance and XC is the capacitive reactance. In this case, XL = 2πfL and XC = 1/(2πfC). Once we have Z, we can calculate the current (I) using Ohm's law: I = V/Z. With the current and the values of the components, we can calculate the voltage drops across each element using Ohm's law again. Finally, we can compare the sum of the voltage drops across the resistor, inductor, and capacitor to the source voltage.
The impedance (Z) = √((R^2) + (((XL - XC)^2))) = √((200^2) + ((2π(250)(0.900) - 1 / 2π(250)(6.00e-6))^2)) = √((200^2) + ((1.41421356237)(250)(0.900 - 1 / (1.57079632679)(250)(6.00e-6))^2)) ≈ 250 Ω
The current (I) = V / Z = 30.0 / 250 = 0.12 A
The voltage drop across the resistor (UR) = I * R = 0.12 * 200 = 24 V
The voltage drop across the inductor (UL) = I * XL = 0.12 * 2π(250)(0.900) = 67.39 V
The voltage drop across the capacitor (UC) = I * XC = 0.12 * (1 / 2π(250)(6.00e-6)) = 0.53 V
The sum of UR + UL + UC = 24 + 67.39 + 0.53 ≈ 92.92 V, which is not equal to the source voltage (V), indicating that there is a phase difference between the current and the voltage.