Question

Explain the generation of LC oscillations in a circuit containing an inductor of inductance L and a capacitor of capacitance C.

Answer 1

Whenever energy is given to a circuit containing a pure inductor of inductance L and a capacitor of capacitance C, the energy oscillates back and forth between the magnetic field of the inductor and the electric field of the capacitor. Thus the electrical oscillations of definite frequency are generated. These oscillations are called LC oscillations.

LC oscillations

Generation of LC oscillations:

1. Let us assume that the capacitor is fully charged with maximum charge Qm at the initial stage. So that the energy store in the Q_m capacitor is maximum and is given by U_E = `"Q"_"m"^2/"2C"`
2. As there is no current in the inductor, the energy stored in it is zero i.e., U_E = 0. Therefore, the total energy is wholly electrical. (Figure (a)).
3. The capacitor now begins to discharge through the inductor that establishes current i in a clockwise direction. This current produces a magnetic field around the inductor and the energy stored in the inductor is given by
   `"U"_"E" = "Li"^2/2`
   As the charge in the capacitor decreases, the energy stored in it also decreases and is given by
   `"U"_"E" = "q"^2/(2"C")` total energy is the sum of electrical and magnetic energies. (Figure (b)).
4. When the charges in the capacitor are exhausted, its energy becomes zero i.e., U_E = 0.
   The energy is fully transferred to the magnetic field of the inductor and its energy is maximum.
5. This maximum energy is given by U_E = `"Li"^2/2` where I_mis the maximum current flowing in the circuit. The total energy is wholly magnetic (Figure (c)).
6. Even though the charge in the capacitor is zero, the current will continue to flow in the same direction because the inductor will not allow it to stop immediately.
7. As a result of this, the capacitor begins to charge in the opposite direction. A part of the energy is transferred from the inductor back to the capacitor. The total energy is the sum of the electrical and magnetic energies. (Figure (d)).
8. (Fig. e) i = o the capacitor becomes fully charged in the opposite direction.
9. (Fig. f) The state of the circuit is similar to the initial state but the difference is that the capacitor is charged in opposite direction.
   ∴ Total energy = U_E + U_B.
10. As already explained, the process repeated in opposite direction (Fig. g and h). Finally alternating current flows in the circuit.
11. This process is repeated again and again to produce LC Oscillations.