Skip to content
CalibratePhysics

13.7

LC Oscillations

AC Circuits · Maharashtra HSC

Theory

13.7 LC Oscillations

A capacitor stores energy in its electric field and an inductor stores energy in its magnetic field. When a fully charged capacitor is connected across an inductor, the charge is transferred to the inductor and a current starts flowing through it.

Because the current is increasing, the magnetic flux of the inductor changes and an induced emf is produced. This self-induced emf opposes the growth of the current. The energy stored on the capacitor decreases and an equivalent amount of energy is stored in the inductor as magnetic field energy. When the capacitor is completely discharged, all the energy that was in the capacitor is stored in the inductor. The inductor then starts charging the capacitor: the current and the flux linked with the inductor begin to decrease, an induced emf is produced which recharges the capacitor in the opposite direction. This charging and discharging repeats, and the energy taken from the source keeps oscillating between the capacitor CC and the inductor LL.

Definition: when a charged capacitor is allowed to discharge through a non-resistive inductor, electrical oscillations of constant amplitude and frequency are produced. These are called LC oscillations.

The Cycle Stage by Stage

  • Stage (a): a capacitor with initial charge q0q_0 at t=0t = 0 is connected to an ideal (zero resistance) inductor. The electrical energy stored in the dielectric medium between the plates is UE=12q02CU_E = \dfrac{1}{2}\dfrac{q_0^2}{C}. There is no current, so the energy stored in the magnetic field of the inductor is zero.
  • Stage (b): the circuit is closed and the capacitor begins to discharge through the inductor, giving rise to a current II. As II increases it builds up a magnetic field around the inductor, and part of the electrical energy is stored as magnetic energy UB=12LI2U_B = \dfrac{1}{2}LI^2.
  • Stage (c): the capacitor is fully discharged and the potential difference across its plates is zero. The current has reached its maximum value I0I_0, and the energy in the magnetic field is 12LI02\dfrac{1}{2}LI_0^2. The entire electrostatic energy of the capacitor has been converted into magnetic field energy.
  • Stage (d): after the discharge is complete the flux linked with the inductor decreases, inducing a current in the same direction as before (Lenz's law). The current persists with decreasing magnitude and charges the capacitor in the opposite direction; the magnetic energy of the inductor changes back into electrostatic energy of the capacitor.
  • Stage (e): the process continues until the capacitor is fully charged with a polarity opposite to that of its initial state, so the entire energy is again 12q02C\dfrac{1}{2}\dfrac{q_0^2}{C} in the electric field.
  • The capacitor now begins to discharge again, sending current in the opposite direction, the energy is once more transferred to the magnetic field, and the whole process repeats in the opposite sense. The circuit eventually returns to its initial state.

Thus the energy of the system continuously surges back and forth between the electric field of the capacitor and the magnetic field of the inductor, producing electrical oscillations of a definite frequency. If there is no loss of energy the amplitude remains constant and the oscillations are undamped.

Why LC Oscillations Are Damped in Practice

  • Every inductor has some resistance. This causes energy loss as heat, the amplitude of the oscillations goes on decreasing and they finally die out.
  • Even if the resistance were zero, the total energy of the system would not remain constant: it is radiated away in the form of electromagnetic waves. The working of radio and TV transmitters is based on such radiation.

Standard Results

UE=12q2C,UB=12LI2U_E = \dfrac{1}{2}\dfrac{q^2}{C}, \qquad U_B = \dfrac{1}{2}LI^2

12q02C=12LI02⇒I0=q0LC=V0CL\dfrac{1}{2}\dfrac{q_0^2}{C} = \dfrac{1}{2}LI_0^2 \qquad \Rightarrow \qquad I_0 = \dfrac{q_0}{\sqrt{LC}} = V_0\sqrt{\dfrac{C}{L}}

f=12πLC,C=14π2f2Lf = \dfrac{1}{2\pi\sqrt{LC}}, \qquad C = \dfrac{1}{4\pi^2f^2L}

q0=CV0q_0 = CV_0 is the initial charge, V0V_0 the initial potential of the capacitor, I0I_0 the maximum current in the coil, and ff the natural frequency of the LC circuit.

Worked Pattern

A 10 μ\muF capacitor charged to 25 V is connected across a pure 100 mH coil: q0=CV0=10×10−6×25=2.5×10−4q_0 = CV_0 = 10\times10^{-6}\times25 = 2.5\times10^{-4} C, and LC=0.1×10−5=10−3\sqrt{LC} = \sqrt{0.1\times10^{-5}} = 10^{-3} s, so I0=2.5×10−4/10−3=0.25I_0 = 2.5\times10^{-4}/10^{-3} = 0.25 A.

For a 101.4 μ\muH inductance to oscillate at 1 MHz, C=14π2f2LC = \dfrac{1}{4\pi^2f^2L} gives C=249.7C = 249.7 pF.