Electrical Oscillator Prepared by; Dr. Rajesh Sharma Assistant Professor Dept of Physics P.G.G.C-11, Chandigarh Email: drrajeshsharma@in.com

Electrical Oscillator A circuit consisting of an inductance (L) and capacitance (C) serves as an electrical oscillator which resembles a “mass-spring” oscillator, where the role of mass is being played by the charge q on the capacitor.

Suppose, a capacitor containing a charge q0 on its plates is connected with an inductor. The capacitor starts discharging through the inductor. (Fig.i) The flow of charge constitutes a current and a magnetic field is setup in the inductor coil L. Fig.ii shows the state of the oscillator when the capacitor is completely discharges and maximum magnetic field is built up in the inductor coil. The variation of the magnetic field linked with the coil sets up a induced emf across it, which in accordance with Lenz’s law is opposite to the potential difference across the capacitor. The emf , in turn charges the capacitor in the opposite direction (Fig.iii). Again, charge q0 is collected on the plates of the capacitor, but the sign of charge is opposite to that it was to start with. (Fig.i) Now, the process of discharging of capacitor begins but the direction of the current is opposite to that of the earlier case. Again a magnetic field is setup across the inductor, but the direction of the magnetic field is opposite to that set up in Fig.ii. The variation of magnetic flux again sets up induced emf which charges the capacitor obtaining the original state (Fig.i). One oscillation is completed and the system is again ready for next oscillation.

Differential Equation for the Electrical Oscillator The induced e.m.f. in the inductor, according to faraday’s law is given by where I is the current in the circuit. since,  The potential drop across the capacitor is , where q is the instantaneous charge on the plates for the capacitor. Applying Kirchoff’s second law, we get Or,

Which can be written as, Or, (1) This is the differential equation for the electrical oscillator. Comparing it with the differential equation for the mechanical oscillator, it is found that the displacement ‘x’ of the mechanical oscillator is represented by the charge ‘q’ , which is the variable physical quantity for the electrical oscillator. Here and hence the frequency of oscillations is given The eq. of the SHM which is also the solution of the Differential equation (2) Where q0 is the amplitude, i.e. the maximum value of the charge on the plates of capacitor

Energy of the capacitor If the charge on the plates of the capacitor is q , then the potential energy across the plates is q/C. Therefore, if we wish to add a small charge dq to the plates, then work done, which is equal to the electric energy stored is given by Therefore, total energy stored on the capacitor, when charge on the plates is q , is given by Or, (2)

Energy of the Inductor The induced emf on the inductor is . So, when the current in the inductor is I , the energy stored is: So, the total energy on the inductor is: But, therefore (4)

Total Energy The total energy of the electrical oscillator or the total EM energy is given by: Or, (5) substituting we get Or, But, . Therefore, . Hence, substituting the value of 1/C, we get Or, (6) This again shows that the total energy of the oscillator is conserved and depends upon L, q0 and w0 as well as C.