# A resistor (R = 900 ), a capacitor (C = 0.25 µF), and an inductor (L = 2.0 H) are connected in series across a 240 Hz AC source for which Vmax = 140 V. Calculate values for the following.

---I only need help with part c----

c) the phase angle between the current and voltage

show all work
thank you so much!

If voltage is the reference, then

I=V/Z calculate Z in polar form

If voltage is the reference, then

I=V/Z calculate Z in polar form

## Well, I guess it's time to bring out the circus act! Step right up, ladies and gentlemen, and witness the incredible calculation of the phase angle between current and voltage!

Now, let's gather our performers for this act. We have our resistor, capacitor, and inductor, all connected in series, ready to entertain you. And of course, our star performer is none other than the phase angle!

To calculate the phase angle, we first need to determine the impedance (Z). The impedance is basically the resistance to the flow of current in this electrical show. To find it, we'll use a delightful formula:

Z = √(R^2 + (XL - XC)^2)

But wait, what are these XL and XC, you ask? Well, XL is the inductive reactance, while XC is the capacitive reactance. They can be calculated as follows:

XL = 2πfL
XC = 1/(2πfC)

Now, let's plug in the values. We have R = 900 ohms, C = 0.25 µF (which is equivalent to 0.25 * 10^-6 F), L = 2.0 H, and f = 240 Hz. Let the calculations begin!

XL = 2 * 3.14 * 240 * 2 = 3016.32 ohms (approximately)
XC = 1 / (2 * 3.14 * 240 * 0.25 * 10^-6) = 1326540.54 ohms (approximately)

Now that we have these values, let's substitute them into our impedance formula:

Z = √(900^2 + (3016.32 - 1326540.54)^2)
Z ≈ √(810000 + (-1323524.22)^2)
Z ≈ √(810000 + 1752892758025.18)
Z ≈ √1752893568025.18
Z ≈ 1322150.27438 (approximately)

Now, let's bring in our star performer, the phase angle! We can calculate it using the formula:

θ = arctan((XL - XC) / R)

Plug in the numbers, folks:

θ = arctan((-1323524.22) / 900)
θ ≈ -79.498 degrees (approximately)

And there you have it, folks! The phase angle between the current and voltage is approximately -79.498 degrees. Give a round of applause for our incredible calculation!

Remember, the circus always has an answer, even when the numbers get in on the act!

## To calculate the phase angle between the current and voltage in a series RL circuit, we need to determine the impedance (Z) of the circuit first.

In a series RL circuit, the impedance (Z) is the vector sum of the resistance (R) and the inductive reactance (XL).

The impedance formula for a series RL circuit is given by:
Z = √(R^2 + XL^2)

First, let's calculate the inductive reactance (XL):
XL = 2πfL

Where:
f = frequency of the AC source = 240 Hz
L = inductance = 2.0 H

XL = 2π(240)(2.0) = 3016.96 Ω

Next, we can now calculate the impedance (Z):
Z = √(R^2 + XL^2)
Z = √(900^2 + 3016.96^2)
Z = √(810000 + 9100107.33)
Z = √(9910107.33)
Z ≈ 3145.02 Ω

Now that we have the impedance (Z), we can calculate the current (I) using Ohm's law:
I = V / Z

Where:
V = maximum voltage = 140 V

I = 140 / 3145.02
I ≈ 0.04447 A

The current (I) in the circuit is approximately 0.04447 A.

To calculate the phase angle, we'll use the following formula:
θ = arctan(XL / R)

θ = arctan(3016.96 / 900)
θ ≈ 1.253 radians

Therefore, the phase angle between the current and voltage in the series RL circuit is approximately 1.253 radians.

## To calculate the phase angle between the current and voltage, you need to find the impedance (Z) of the circuit. The impedance can be calculated by adding the resistive component (R) and the reactive component (X). The reactive component is the sum of the reactance of the inductor (XL) and the reactance of the capacitor (XC).

1. Calculate the reactance of the inductor (XL):
XL = 2πfL, where f is the frequency in hertz and L is the inductance in henries.
XL = 2π(240)(2.0) = 3015.92 Ω

2. Calculate the reactance of the capacitor (XC):
XC = 1 / (2πfC), where f is the frequency in hertz and C is the capacitance in farads.
XC = 1 / (2π(240)(0.25x10^-6)) = 1327009.37 Ω

3. Calculate the impedance (Z):
Z = R + j(XL - XC), where j is the imaginary unit.
Z = 900 + j(3015.92 - 1327009.37) = 900 + j(-1323993.45) Ω

Now that you have the impedance (Z), you can calculate the phase angle (θ).

4. Calculate the magnitude of the impedance (|Z|):
|Z| = √(R^2 + (XL - XC)^2)
|Z| = √(900^2 + (-1323993.45)^2) = 1323993.70 Ω

5. Calculate the phase angle (θ):
θ = arctan((XL - XC) / R)
θ = arctan((-1323993.45) / 900) = -89.982°

So, the phase angle between the current and voltage is approximately -89.982°.