Question

Consider the LC circuit. If one needs to tune this circuit to a frequency of 84 kHz, and the capacitor has a capacitance C = 3.0 mF, what inductance L is needed?

Answer 1

The resonant frequency of an LC circuit occurs when the capacitive reactance equals the inductive reactance:

`X_C = X_L`

Substitute `X_C = 1/(2pifC)`:

`1/(2pifC) = X_L`

Substitute `X_L = 2pifL`:

`1/(2pifC) = 2pifL`

Solve for L:

`L = 1/(4pi^2f^2C)`

Substitute `f = 84xx10^3" Hz"`:

`L = 1/(4pi^2(84xx10^3" Hz")^2C)`

NOTE: There is an error in the wording of this problem; it specifies that `C = 3.0" mF"` but the given answer implies that the capacitance must be `C = 3.0" "mu"F"`

Substitute `C = 3.0xx10^-6" F"`:

`L = 1/(4pi^2(84xx10^3" Hz")^2(3.0xx10^-6" F"))`

`L ~~ 1.2xx10^-6" H"`