Question

At what frequency does the inductive reactance of a 42.0 µH inductor equal the capacitive reactance of a 42.0 µF capacitor?
kHz

Answer 1

reactance magnitude = V/i

for inductance
V = L di/dt
if i = a sin wt
di/dt = a w cos wt
and
V = L a w cos wt
magnitude of V/i = Lw = 2 pif L
here L = 42 * 10^-6
so
reactance = 2 pi f (42*10^-6)

for capacitor
C = q/V
V = q/C = (1/C) integral i dt
if i = a sin w t
V = (1/C) (a/w)(-cos wt)
magnitude of reactance =
1/(wC) = 1/(2 pi f C)
= 1/(2 pi f (42*10^-6))

so
1/(2 pi f (42*10^-6)) = 2 pi f (42*10^-6)

1 =( 2 pi f)^2 (42^2) 10^-12

(2 pi f)^2 = 10^12/42^2
2 pi f = 10^6/42 = 2.38*10^6
f = .379 * 10^6
= 379,000 Hz

Answer 2

2 pi f = 10^6/42 = 2.38*10^4
f = .379 * 10^4
= 3790 Hz