Question

The lc circuit of a radar transmitter oscillates at 9.00 ghz. (a) what inductance is required for the circuit to resonate at this frequency if its capacitance is 2.00 pf? (b) what is the inductive reactance of the circuit at this frequency?

Answer 1

(a) The resonant frequency of the circuit is

`f=9.00 GHz = 9.00 \cdot 10^9 Hz`

The relationship between the resonant frequency of a circuit, the inductance L and the capacity C is

`f= (1)/(2 \pi √(LC))`

Since the capacitance in the circuit is
`C=2.00 pF= 2.00 \cdot 10^(-12) F`, we can use the previous formula to find the inductance L:

`L= (1)/((2 \pi f)^2 C) = (1)/((2 \pi 9.00 \cdot 10^9 Hz)^2(2.00 \cdot 10^(-12) F))= 1.57 \cdot 10^(-10) H`

b) The inductive reactance of the circuit is given by:

`X_L = 2 \pi f L`
where f is the frequency of the circuit. The frequency is still the same of the previous part of the exercise, so the inductive reactance is:

`X_L = 2 \pi (9.00 \cdot 10^9 Hz)(1.57 \cdot 10^(-10)H)=8.87 \Omega`