Name: RC Circuit and Charging Capacitor
Uploaded: 2019-07-19
Duration: 13 min 36 s
Description: In this video, we will discuss the physics of charging the capacitor connected in a circuit and derive the expression and relation between the parameters inv...

Question

A charged capacitor is connected across an inductor with a wire having practically no resistance. Show that the discharge of the capacitor is oscillatory. Find the frequency of oscillations.

Accepted Answer

Both capacitors and inductors store energy, in their electric and magnetic fields, respectively. A circuit containing both an inductor $(L)$ and a capacitor $(C)$ can oscillate without a source of emf by shifting the energy stored in the circuit between the electric and magnetic fields.

If the capacitor contains a charge $q_{o}$ before the switch is closed, then all the energy of the circuit is initially stored in the electric field of the capacitor. This energy is $U_{C} = \frac{1}{2} \frac{q^{2}}{C}$ .

When the switch is closed, the capacitor begins to discharge, producing a current in the circuit. The current, in turn, creates a magnetic field in the inductor. The net effect of this process is a transfer of energy from the capacitor, with its diminishing electric field, to the inductor, with its increasing magnetic field.

Energy in inductor $U_{L} = \frac{1}{2} L I^{2}$

Since there is no resistance in the circuit, no energy is lost through Joule heating; thus, the maximum energy stored in the capacitor is equal to the maximum energy stored at a later time in the inductor.

$∴ \frac{1}{2} \frac{{q_{o}}^{2}}{C} = \frac{1}{2} L I^{2}$

In this circuit total energy stored by capacitor is equal to inductor due to no loss of energy. So circuit will oscillate periodically.

Angular frequency $ω$ $= \sqrt{\frac{1}{L C}}$ .

A charged capacitor is connected across an inductor with a wire having practically no resistance. Show that the discharge of the capacitor is oscillatory. Find the frequency of oscillations.

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