Final answer:
The impedance for each component at 60 Hz can be calculated using the formulas ZR = R, ZL = jωL, and ZC = 1/jωC. The total impedance in series is found by adding the individual impedances.
Step-by-step explanation:
Impedance is the total opposition to the flow of alternating current (AC) in an electrical circuit. It is represented by the symbol Z and is a combination of resistance (R), inductive reactance (XL), and capacitive reactance (XC) in a circuit.
To find the impedance for each component at 60 Hz, we can use the formulas:
ZR = R (the resistance)
ZL = jωL (the inductive reactance)
ZC = 1/jωC (the capacitive reactance)
Where j is the imaginary unit (√-1), ω is the angular frequency (2πf), L is the inductance, and C is the capacitance.
Once we have the individual impedances, we can find the total impedance in series by adding them together: Ztotal = ZR + ZL + ZC.
For the given components:
a. 1000 Ω resistor: ZR = 1000 Ω
b. 500 mH inductor: ZL = jωL = j(2πf)L = j(2π(60))(0.5) = 0.188i Ω
c. 1 μF capacitor: ZC = 1/jωC = 1/(j(2πf)C) = 1/(j(2π(60))(1x10^-6)) = -2653.98i Ω
To find the total impedance in series, we add the individual impedances: Ztotal = 1000 Ω + 0.188i Ω - 2653.98i Ω = (1000 - 2653.792)i Ω = -1653.792i Ω