Answer:
"√2f
"An inductor’s impedance is twice that of a capacitor at a specific frequency (f).
Both components are connected in series.
We need to find the new resonant frequency.
Impedance relationship
The inductive reactance XL and capacitive reactance XC are related by:
XL = 2XC
Substituting their formulas:
$omega L = 2 cdot frac{1}{omega C}$
where
$ omega = 2pi f $ is the angular frequency.
Step 2: Solve for angular frequency $$ omega $$
Rearranging the equation:
$omega^2 L = frac{2}{C}$
$omega = sqrt{frac{2}{LC}}$
Step 3: Resonant frequency formula
The resonant angular frequency is related to the resonant frequency ($ f_text{new}$) by:
$omega = 2pi f_text{new}$
Substitute $ omega = sqrt{frac{2}{LC}}$:
$2pi f_text{new} = sqrt{frac{2}{LC}}$
Solve for $ f_text{new}$:
$f_text{new} = frac{sqrt{2}}{2pi sqrt{LC}}$
Step 4: Compare with the original resonant frequency
The original resonant frequency ($ f_text{original}$) is given by:
$f_text{original} = frac{1}{2pi sqrt{LC}}$
Now compare:
$f_text{new} = sqrt{2} cdot f_text{original}$
The new resonant frequency is:
$f_text{new} = sqrt{2}f$