Solving for unknown with square root as a term

Question

I should be able to crank this out easily! Unfortunately not. So need some basic help here...

I'm trying to determine parasitic inductance in an unknown inductor based on change of ringing frequency. By inserting a capacitance across the low-side switch (nChannel FET) when the ringing frequency is cut in half, the parasitic capacitance is equal to 1/3 of the capacitance value.

To determine the inductance I have the formula: $$ \ f = 0.5 \pi \sqrt LC $$

...which I believe is a derivation of: $$ \ Fr = 1 / (2\pi \sqrt LC) $$

To solve for L I believe it's first necessary to remove the square root by squaring both sides? But what of $$ 0.5 \pi $$

My known terms are f=9.3MHz, C=1000pf. How to solve for L?

Answer 1

Isolate $\sqrt L$: $$\sqrt L = \dfrac f{.5\pi \cdot C} = \frac{2f}{\pi C}\tag {$\frac 1{\frac 12} = 2$}$$

Now square each side square each side of the equation to get

$$(\sqrt L)^2= \left(\dfrac{2f}{\pi C}\right)^2$$

$$L = \left(\frac {4f^2}{(\pi C)^2}\right)$$

Answer 2

Assuming both $L$ and $C$ are supposed to be under the square root sign, just square the equation. The $0.5\pi$ when squared gives $0.25\pi^2$, so $$f=0.5\pi \sqrt {LC}\\f^2=0.25\pi^2LC\\L=\frac {f^2}{0.25\pi^2C}=\frac {4f^2}{\pi^2C}$$