Correct Answer - Option 3 :
\(\frac{{\rm{R}}}{2}\sqrt {\frac{{\rm{C}}}{{\rm{L}}}}\)
The characteristic equation of a series RLC circuit is given by,
\({\rm{C}}.{\rm{E}} = {{\rm{s}}^2} + \frac{{\rm{R}}}{{\rm{L}}}{\rm{s}} + \frac{1}{{{\rm{LC}}}}\)
We have \({{\rm{\omega }}_{\rm{n}}} = \frac{1}{{\sqrt {{\rm{LC}}} }}\)
And \(2{\rm{\zeta }}{{\rm{\omega }}_{\rm{n}}} = \frac{{\rm{R}}}{{\rm{L}}}\)
\(\begin{array}{l}
\Rightarrow {{\rm{\omega }}_{\rm{n}}} = \frac{{\rm{R}}}{{\rm{L}}} \times \frac{1}{2} \times \sqrt {{\rm{LC}}} \\
\Rightarrow {{\rm{\omega }}_{\rm{n}}} = \frac{{\rm{R}}}{2}\sqrt {\frac{{\rm{C}}}{{\rm{L}}}}
\end{array}\)