Answer:
To find the resonance frequency of a series RLC circuit, we can use the formula:
f = 1 / (2π√(LC))
Where:
f = Resonance frequency
L = Inductance in henries
C = Capacitance in farads
π = 3.14159...
In this case, we are given the resistance and inductive impedance of the coil, but not its inductance. However, we know that the inductive impedance of a coil is given by:
XL = 2πfL
Where:
XL = Inductive impedance
f = Frequency
L = Inductance in henries
π = 3.14159...
At resonance, the inductive impedance of the coil will equal the capacitive reactance of the capacitor:
XL = XC
Where:
XC = Capacitive reactance
XC = 1 / (2πfC)
Substituting XL and XC into the equation above, we get:
2πfL = 1 / (2πfC)
Simplifying this equation, we get:
f = 1 / (2π√(LC))
Where:
L = XL / (2πf) = 65Ω / (2πf)
C = 49µF = 49 × 10^-6F
Substituting these values into the resonance frequency equation, we get:
f = 1 / (2π√(65Ω/(2πf) × 49 × 10^-6F))
Simplifying this equation, we get:
f = 1 / (2π√((3.385 × 10^-6)/f))
Multiplying both sides by 2π√((3.385 × 10^-6)/f), we get:
2π√((3.385 × 10^-6)/f) × f = 1
Squaring both sides, we get:
4π^2(3.385 × 10^-6)/f = 1
Solving for f, we get:
f = √((4π^2 × 3.385 × 10^-6))
f ≈ 1369 Hz.
Therefore, the resonance frequency of the circuit is approximately 1369 Hz.