An RLC oscillator circuit contains a 63.3-Ω resistor and a 1.67-mH inductor. What capacitance is necessary for the time constant of the circuit (the 1/e value) to be equal to the oscillation period? (You may enter your calculation using scientific notation.) F
Question
An RLC oscillator circuit contains a 63.3-Ω resistor and a 1.67-mH inductor. What capacitance is necessary for the time constant of the circuit (the 1/e value) to be equal to the oscillation period? (You may enter your calculation using scientific notation.) F
Solution
The time constant (τ) of an RLC circuit is given by the formula:
τ = L/R
where L is the inductance and R is the resistance.
The oscillation period (T) of an RLC circuit is given by the formula:
T = 2π√(LC)
where L is the inductance and C is the capacitance.
We are given that τ = T, so we can set the two equations equal to each other and solve for C:
L/R = 2π√(LC)
Squaring both sides to get rid of the square root gives:
(L/R)^2 = 4π^2LC
Rearranging to solve for C gives:
C = (L/R^2) * (1/4π^2)
Substituting the given values:
C = (1.67 x 10^-3 H / (63.3 Ω)^2) * (1/4π^2)
C = 1.67 x 10^-7 F
So, the necessary capacitance for the time constant of the circuit to be equal to the oscillation period is approximately 1.67 x 10^-7 F.
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