Analogies between LC oscillations and simple harmonic oscillations

**Analogies between**** ****LC oscillations and simple harmonic oscillations**

The electromagnetic
oscillations of *LC* system can be compared with the mechanical oscillations
of a spring-mass system.

There are two forms of
energy involved in *LC* oscillations. One is electrical energy

Likewise, the mechanical
energy of the spring-mass system exists in two forms; the potential energy of
the compressed or extended spring and the kinetic energy of the mass. The Table
4.3 lists these two pairs of energy.

By examining the Table
4.3, the analogies between the various quantities can be understood and these
correspondences are given in the Table 4.4.

The angular frequency of
oscillations of a spring-mass is given by (Refer equation 10.22 of section
10.4.1 of XI physics text book).

From Table 4.4, *k*
→ ^{1}**/*** _{C}* and

The mechanical energy of
the spring-mass system is given by

The energy *E*
remains constant for varying values of *x* and *v*. Differentiating *E*
with respect to time, we get

This is the differential
equation of the oscillations of the spring-mass system. The general solution of
equation (4.68) is of the form

where *X _{m}*
is the maximum value of

Similarly, the
electromagnetic energy of the *LC* system is given by

The general solution of
equation (4.71) is of the form

where *Q _{m}*
is the maximum value of

The current flowing in
the *LC* circuit is obtained by differentiating *q(t)* with respect
to time.

The equation (4.73)
clearly shows that current varies as a function of time *t*. In fact, it
is a sinusoidally varying alternating current with angular frequency *ω*.

By differentiating
equation (4.72) twice, we get

Substituting equations
(4.72) and (4.74) in equation (4.71), we obtain

This equation is the same as that obtained from qualitative analogy.

The electrical energy of
the LC oscillator is

If the two energies are
plotted with an assumption of *f* =
0 , we obtain Figure 4.58.

From the graph, it can
be noted that

(i) At any instant U_{E}+U_{B}=Q_{m}^{2}/2C
= constant

(ii) The maximum values of U_{E}
and U_{B} are both Q_{m}/2C

(iii) When U_{E} is Maximum,
U_{B} is zero and vice versa.

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12th Physics : Electromagnetic Induction and Alternating Current : Analogies between LC oscillations and simple harmonic oscillations |

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