Question

A series RCL circuit has a resonant frequency of 1500 Hz. When operating at a frequency other than 1500 Hz, the circuit has a capacitive reactance of 5.0 Ù and an inductive reactance of 30.0 Ù. What are the values of (a) L and (b) C?

I know I should be using these 2 equations:
Xc= 1/(2*pi*f*C)
Xl= 2*pi*f*L

Since each have 2*pi*f in them, with rearranging and substituting, you get:

C= L/(Xl*Xc)

However, with this it leaves the 2 unknowns, and I'm not sure what other equations I could use?

Answer 1

f=1500 Hz, X(C)=5Ω, X(L)=30 Ω

X(C)= 1/ωC,…..(1)
X(L)=ωL, ……..(2)
Multiply (1) by (2):
X(C) •X(L)= ωL/ ωC =L/C.
C=L/X(C) •X(L) …..(3)
Resonant frequency is
f=1/2π√(LC) =>
C=1/4•π²•f²•L ….(4)
Equate (3) and (4)
L/X(C) •X(L) =1/4•π²•f²•L,
L=sqrt{X(C) •X(L}/2•π•f=
=sqrt(5•30)/2π•1500=1.3•10^-3 H=
=1.3 mH,
C= L/X(C) •X(L)=
=1.3•10^-3/5•30=8.67•10^-6 =F=
= 8.67 μF