A series RLC circuit has a capacitor with a capacitance of 45.0 μF , an inductor with an inductance of 1.40 H and a resistor with a resistance of 59.0 Ω. The circuit has a rms current of 4.70 A when the frequency is 98.0 Hz. What is εrms?
Question
A series RLC circuit has a capacitor with a capacitance of 45.0 μF , an inductor with an inductance of 1.40 H and a resistor with a resistance of 59.0 Ω. The circuit has a rms current of 4.70 A when the frequency is 98.0 Hz. What is εrms?
Solution
The root mean square (rms) voltage (εrms) in a series RLC circuit can be calculated using the formula:
εrms = Irms * Z
where:
- Irms is the rms current,
- Z is the impedance of the circuit.
The impedance (Z) of a series RLC circuit is given by the formula:
Z = sqrt(R^2 + (XL - XC)^2)
where:
- R is the resistance,
- XL is the inductive reactance, given by the formula XL = 2πfL,
- XC is the capacitive reactance, given by the formula XC = 1/(2πfC),
- f is the frequency,
- L is the inductance,
- C is the capacitance.
Let's calculate the values:
XL = 2π * 98 Hz * 1.40 H = 868.23 Ω XC = 1/(2π * 98 Hz * 45.0 μF) = 34.56 Ω
Now, we can calculate the impedance:
Z = sqrt((59.0 Ω)^2 + (868.23 Ω - 34.56 Ω)^2) = 834.67 Ω
Finally, we can calculate the rms voltage:
εrms = 4.70 A * 834.67 Ω = 3922.35 V
So, the rms voltage in the circuit is approximately 3922.35 V.
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