You have a 200-Ω resistor, a 0.400-H inductor, and a 6.00-μF capacitor. Suppose you take the resistor and inductor and make a series circuit with a voltage source that has voltage amplitude 30.0 V and an angular frequency of 250 rad/s. (a) What is the impedance of the circuit?

Verified step by step guidance

1. Calculate the inductive reactance (X_L) using the formula X_L = \omega L, where \omega is the angular frequency and L is the inductance of the inductor.
2. Identify the resistance (R) of the resistor, which is given directly in the problem.
3. Since the circuit is a series RL circuit, calculate the total impedance (Z) using the formula Z = \sqrt{R^2 + X_L^2}. This formula arises from the Pythagorean theorem, as impedance in an RL circuit is the vector sum of resistance and inductive reactance.
4. Substitute the values of R and X_L calculated from the previous steps into the impedance formula to find the magnitude of the total impedance.
5. The phase angle (\phi) between the voltage and the current can be found using \tan^{-1}(\frac{X_L}{R}), but this is not required to find the impedance.

Key Concepts

Impedance

Impedance is a measure of how much a circuit resists the flow of alternating current (AC) and is represented as a complex number. It combines both resistance (R) and reactance (X), where reactance arises from inductors and capacitors. The total impedance in a series circuit can be calculated using the formula Z = √(R² + X²), where X is the net reactance.

Reactance

Reactance is the opposition to the change in current or voltage in an AC circuit, caused by inductors and capacitors. Inductive reactance (XL) increases with frequency and is calculated as XL = ωL, where ω is the angular frequency and L is the inductance. Capacitive reactance (XC) decreases with frequency and is given by XC = 1/(ωC), where C is the capacitance.

Series Circuit

A series circuit is a type of electrical circuit in which components are connected end-to-end, so the same current flows through each component. In a series circuit, the total impedance is the sum of the individual resistances and reactances. This configuration affects the overall voltage and current in the circuit, making it essential to understand when analyzing AC circuits.