Question

If a LC circuit is considered analogous to a harmonically oscillating spring block system, which energy of the LC circuit would be analogous to potential energy and which one analogous to kinetic energy?

Answer 1

When a charged capacitor C having an initial charge q0 is discharged through an inductance L, the charge and current in the circuit starts oscillating simple harmonically. If the resistance of the circuit is zero, no energy is dissipated as heat. We also assume an idealized situation in which energy is not radiated away from the circuit. The total energy associated with the circuit is constant.

The oscillation of the LC circuit is electromagnetic analogous to the mechanical oscillation of a block-spring system.

The total energy of the system remains conserved.

`1/2 CV^2 + 1/2 LI^2` = constant = `1/2 CV_0^2 = 1/2 LI^2`

At t = `(3T)/4`, capacitor again discharges completely i = i_max

At `t = (T)/2`, block again reaches its mean position and its velocity becomes maximum

Comparison of oscillation of a mass-spring system and an LC circuit

Mass spring system	v/s	LC circuit
Displacement `(x)`		Charge `(q)`
Velocity `(v)`		Current `(i)`
Acceleration `(a)`		Rate of change of current `((di)/(dt))`
Mass `(m)` [Inertia]		Inductance `(L)` [Inertia of electricity]
Momentum `(p = mv)`		Magnetic flux `(phi = Li)`
Retarding force `(-m (dv)/(dt))`		Self induced emf `(-L (di)/(dt))`

 

Equation of free oscillations: `(d^2x)/(dt^2) = - ω^2x`; where ω = `sqrt(K/m)`		Equation of free oscillations: `(d^2q)/(dt^2) = - (1/(LC)).q`; where ω2 = `1/(LC)`
Force constant K		Capacitance C
Kinetic energy = `1/2 mv^2`		Magnetic energy = `1/2 Li^2`
Elastic potential energy = `1/2 Kx^2`		Electrical potential energy = `1/2 q^2/C`

If we consider an L-C circuit analogous to a harmonically oscillating spring block system. The electrostatic energy `1/2 CV^2` is analogous to potential energy and energy associated with moving charges (current) that is magnetic energy `(1/2 LI^2)` is analogous to kinetic energy.