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7.Alternating Current
medium
An $LC$ circuit consists of a capacitor and a coil with a large number of turns. Suppose all the linear dimensions of all elements of the circuit are increased by a factor of $2$ while keeping the number of turns on the coil constant. How much does the resonant frequency of the circuit change?
A
becomes two times
B
becomes half
C
becomes one fourth
D
becomes four times
Solution
$\omega=\frac{1}{\sqrt{\mathrm{LC}}}$
$ = \frac{1}{{\sqrt {{\mu _0}{{\rm{n}}^2}{\rm{A}}\ell \times \frac{{{ \in _0}{{\rm{A}}_1}}}{{\rm{d}}}} }} = \frac{1}{{\sqrt {\frac{{{\mu _0}{{\rm{N}}^2}}}{\ell }{\rm{A}} \times \frac{{{ \in _0}{{\rm{A}}_1}}}{{\rm{d}}}} }}$
$\Rightarrow \omega=\frac{\omega_{0}}{2}$
Standard 12
Physics
