7.8 LC O SCILLATIONS

. LC O SCILLATIONS We know that a capacitor and an inductor can store electrical and magnetic energy, respectively. When a capacitor (initially charged) is connected to an inductor, the charge on the capacitor and the current in the circuit exhibit the phenomenon of electrical oscillations similar to oscillations in mechanical systems (Chapter , Class XI). Let a capacitor be charged q m (at t = ) and connected to an inductor as shown in Fig. . . The moment the circuit is completed, the charge on the capacitor starts decreasing, giving rise to current in the circuit. Let q and i be the charge and current in the circuit at time t . Since d i /d t is positive, the induced emf in L will have polarity as shown, i.e., v b < v a . According to Kirchhoff’s loop rule, i L ( . ) i = – (d q /d t ) in the present case (as q decreases, i increases). Therefore, Eq. ( . ) becomes: LC ( . ) This equation has the form x x for a simple harmonic oscillator. The charge, therefore, oscillates with a natural frequency LC ( . ) and varies sinusoidally with time as ( cos φ ( . ) where q m is the maximum value of q and φ is a phase constant. Since q = q m at t = , we have cos φ = or φ = . Therefore, in the present case, FIGURE . At the instant shown, the current is increasing so the polarity of induced emf in the inductor is as shown. cos( ( . ) The current i ⎛ ⎞ = − ⎜ ⎟ ⎝ ⎠ is given by sin( i i ( . ) where i Let us now try to visualise how this oscillation takes place in the circuit. Figure . (a) shows a capacitor with initial charge q m connected to an ideal inductor. The electrical energy stored in the charged capacitor is E U . Since, there is no current in the circuit, energy in the inductor is zero. Thus, the total energy of LC circuit is, E U U FIGURE