An RLC series circuit has a 2.60 Ω resistor, a 110 µH inductor, and a 70.0 µF capacitor.
(a) Find the circuit's impedance (in Ω) at 120 Hz.
(b) Find the circuit's impedance (in Ω) at 5.00 kHz.
(c) If the voltage source has Vrms = 5.60 V, what is Irms (in A) at each frequency?
(d) What is the resonant frequency (in kHz) of the circuit?
(e) What is Irms (in A) at resonance?
w = 2 pi f
impedance = sqrt [ R^2 + (wL-1/wC)^2 ]
if f = 120 Hz, w = 2 pi f = 754
impedance = sqrt [2.6^2 + (754*110*10^-6 - 1/754*70*10^-6 )^2]
repeat for new f
i = V/impedance = 5.6/impedance
at resonance w = 2 pi f = 1/sqrt(LC)
then
impedance = sqt [ R^2 + ( sqrt L/C - sqrt L/C)^2) ] = R
so i = V/R
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Carol is most likely to reject the ___________ theory of aging because she pictures the body as a mach that stops working properly as the parts wear out.
Which of the following statements regarding adolescent suicide is not true ?
To find the impedance of an RLC series circuit, we need to use the following formula:
Z = √(R^2 + (XL - XC)^2)
Where:
Z is the impedance in ohms (Ω)
R is the resistance in ohms (Ω)
XL is the inductive reactance in ohms (Ω)
XC is the capacitive reactance in ohms (Ω)
(a) Impedance at 120 Hz:
To find the impedance at 120 Hz, substitute the given values into the formula.
R = 2.60 Ω
L = 110 µH = 0.110 mH (since 1 H = 10^6 µH)
C = 70.0 µF = 0.070 mF (since 1 F = 10^6 µF)
f = 120 Hz
First, calculate the inductive and capacitive reactances:
XL = 2πfL = 2π(120 Hz)(0.110 mH) = 2π(0.132) = 0.827 Ω
XC = 1/(2πfC) = 1/(2π(120 Hz)(0.070 mF)) = 1/(2π(0.084)) = 0.475 Ω
Now, substitute the values into the impedance formula:
Z = √(R^2 + (XL - XC)^2)
= √(2.60^2 + (0.827 - 0.475)^2)
≈ √(6.76 + 0.138)
≈ √6.898
≈ 2.625 Ω
Therefore, the impedance of the circuit at 120 Hz is approximately 2.625 Ω.
(b) Impedance at 5.00 kHz:
Repeat the same process as above, but with the frequency changed to 5.00 kHz.
f = 5.00 kHz = 5000 Hz
XL = 2πfL = 2π(5000 Hz)(0.110 mH) = 2π(0.550) = 3.456 Ω
XC = 1/(2πfC) = 1/(2π(5000 Hz)(0.070 mF)) = 1/(2π(0.350)) = 0.905 Ω
Z = √(R^2 + (XL - XC)^2)
= √(2.60^2 + (3.456 - 0.905)^2)
≈ √(6.76 + 8.717)
≈ √15.477
≈ 3.93 Ω
Therefore, the impedance of the circuit at 5.00 kHz is approximately 3.93 Ω.
(c) Irms at each frequency:
Now that we know the impedance at both frequencies, we can calculate the current (Irms) using Ohm's Law (I = V/Z), where I is the current, V is the voltage, and Z is the impedance.
Vrms = 5.60 V
At 120 Hz:
Irms = Vrms / Z = 5.60 V / 2.625 Ω ≈ 2.133 A
At 5.00 kHz:
Irms = Vrms / Z = 5.60 V / 3.93 Ω ≈ 1.422 A
Therefore, the current at 120 Hz is approximately 2.133 A, and the current at 5.00 kHz is approximately 1.422 A.
(d) Resonant frequency:
The resonant frequency (fr) of an RLC series circuit is given by the formula:
fr = 1 / (2π√(LC))
Where:
L is the inductance in henries (H)
C is the capacitance in farads (F)
Given values:
L = 110 µH = 0.110 mH
C = 70.0 µF = 0.070 mF
fr = 1 / (2π√(0.110 mH x 0.070 mF))
= 1 / (2π√(0.0077 H ⋅ F))
= 1 / (2π√(0.0077))
= 1 / (2π ∗ 0.088)
≈ 1 / 0.55
≈ 1.82 kHz
Therefore, the resonant frequency of the circuit is approximately 1.82 kHz.
(e) Irms at resonance:
To find the current at resonance, substitute the resonant frequency into the impedance formula:
Z = √(R^2 + (XL - XC)^2)
= √(2.60^2 + (XL - XC)^2)
= √(2.60^2 + (0.827 - 0.475)^2)
≈ √(6.76 + 0.138)
≈ √6.898
≈ 2.625 Ω
Irms = Vrms / Z = 5.60 V / 2.625 Ω ≈ 2.133 A
Therefore, the current at resonance is approximately 2.133 A.