Derive an expression for resonant frequency of series circuit containing inductor, capacitor and resistor.

Explanation

Consider a pure capacitor of capacitance .C. connected to an AC source as shown in fig.
The applied instantaneous voltage is given by `V = V_0 sin omega t ` ....(1)
The charge on the capacitor at time t. is q= CV
`q = CV_0 sin omega t ` ....(2)
The instantaneous current in the circuit is `I = (dq)/(dt)` ...(3)
Equation (2) in (3), we get
`I = (d)/(dt) (CV_0 sin omegat )`
`I=CV_0 ( cos omega t) omega`
`I = (V_0 omega C) cos omega t `
`I = (V_0)/( (1)/(omega C)) sin (omega t +pi/2) " " (because cos omega t = sin (omega t +pi/2) )`
`I = I_0 sin (omega t + pi/2)`
Where `I_0 = (V_0)/((1)/(omega C))` peak value of current.
From equations (1) and (4) it is clear that the current leads the applied voltage by `pi//2 ` or `90^@`