Answer to Question #314308 in Optics for Parker

Question

An inductor, a capacitor, and a resistor are all connected in series across an ac source.

If the resistance, inductance, and capacitance are all doubled, by what factor does each of the

following quantities change

Answer 1

LCR series combination

Impedance

$Z=\sqrt{R^2+(X_c-X_L)}$

$X_L=X_c$

Resonance frequency

$wL=\frac{1}{wC}$

$w=\frac{1}{\sqrt{LC}}$

L=2L

C=2C

$w'=\frac{1}{\sqrt{4LC}}$

$\frac{w'}{w}=\frac{1}{2}$

$w'=\frac{w}{2}$

Inductive reactance

$X_L=wL$

$X'_L={2wL}\\L'=2L$

$\frac{X'_L}{X_L}=\frac{2w_L}{wL}$

$X'_L=2X_L$

Capacitive reactance

$X_c=\frac{1}{wC}$

C'=2C

$X'_c=\frac{1}{2wC}$

$\frac{X'_c}{X_c}=\frac{1}{2}$

$X'_C=\frac{X_c}{2}$