An oscillating circuit consists of a capacitor of capacitance `C` and a solenoid with inductance `L_(1)`. The solenoid is inductively connected with a short`-` circuited coil having an inductance `L_(2)` and a negligible active resistance. Their mutual inductance coefficient is eqaul to `L_(12)`. Find the natural frequency of the given oscillating ciruit.

Explanation

`L_(1)(dI_(1))/( dt)+(intI_(1)dt)/(C)=-L_(12)(dI_(2))/( dt )`
` L_(2)(dI_(2))/( dt)=- L_(12)(dI_(2))/(dt)`
from the second equation
`L_(2) I_(2) =- L_(12)I_(1)`
Then `(L_(2)-(L_(12)^(2))/( L_(2)))ddot(I_(1))+(I_(1))/(C)=0`
Thus the current oscillates with frequency
`omega=(1)/(sqrt(C(L_(1)-(L_(12)^(2))/(L_(2)))))`