Question

To achieve a resonant frequency of 1.00 GHz using an 8.00 nH inductor, what capacitance is required?

a) 2.00 pF
b) 1.00 pF
c) 0.50 pF
d) 0.25 pF

Answer 1

To achieve a resonant frequency of 1.00 GHz with an 8.00 nH inductor, a capacitance of 0.25 pF is required. The resonant frequency formula is used to calculate the necessary capacitance for the LC circuit.

Step-by-step explanation:

To find the capacitance needed to produce a resonant frequency of 1.00 GHz with an 8.00 nH inductor, we can use the resonant frequency formula for an LC circuit: f = 1 / (2 π √(LC)), where f is the frequency, L is the inductance, and C is the capacitance. Solving for the capacitance gives us:

C = 1 / (4 π² f² L)

Plugging in the values results in:

C = 1 / (4 π² (1.00 GHz)² (8.00 nH))

C = 1 / (4 π² (1.00 x 10⁹ Hz)² (8.00 x 10⁻⁹ H))

C ≈ 0.25 pF

Therefore, to achieve a resonant frequency of 1.00 GHz using an 8.00 nH inductor, a capacitance of 0.25 pF is required, which corresponds to option d).