All Problems Electric Oscillations
Problem 4.111
An oscillating circuit incorporates a leaking capacitor. Its capacitance is equal to \(C\) and active resistance to \(R .\) The coil inductance is \(L .\) The resistance of the coil and the wires is negligible. Find: (a) the damped oscillation frequency of such a circuit; (b) its quality factor.
4.111. (a) \(\omega=\sqrt{\frac{1}{L C}-\frac{1}{4 R^{2} C^{2}}}\); (b) \(Q=\frac{1}{2} \sqrt{\frac{4 R^{2} C}{L}-1}\). When solving the problem, it should be taken into account that \(d q / d t=I-I^{\prime},\) where \(q\) is the charge of the capacitor, \(I\) is the current in the coil winding, \(I^{\prime}\) is the leakage current \(\left(I^{\prime}=V / R\right)\).