In a spring-mass system the total mechanical energy is represented as: $TE= 1/2kx^2 + 1/2mv^2$ $k$ being the spring constant and $x$ being the displacement.

In L.C Oscillations we find the total electromagnetic energy to be: $TE=1/2 Q^2/LC + 1/2 Li^2$ $L$ being the inductance of the inductor and $C$ is the Capacitance of the capacitor both in resonance which is why the L.C oscillations take place.

My book says that the electrical energy i.e $1/2 Q^2/LC$ "resembles" P.E of spring which is $1/2kx^2$. I understand that we come across numerous situations in physics that are represented by the same or similar mathematical equations. I was wondering if this has some physical implications/significance or is it just an analogy.


2 Answers 2


You can expect this form universally for any small perturbation to a stable equilibrium. The reason is that any smooth energy minimum, regardless of its shape, looks like a parabola up close. This can be shown from a Taylor series expansion of the energy $U$ around the equilibrium point:

$$U(0+x)=U(0)+U^{\prime}(0)x+\frac{1}{2}U^{\prime\prime}(0)x^2+\cdots\approx \frac{1}{2}U^{\prime\prime}(0)x^2,$$

where the prime notation indicates derivatives, where $U(0)$ is taken as zero (an arbitrary reference point), where $U^\prime(0)=0$ because we're describing perturbations around a minimum, and where I've dropped all subsequent terms as negligible. Here, $U^{\prime\prime}(0)$ represents a single number: the curvature of the energy well at its minimum. Thus, $$U(x)\propto\frac{1}{2}x^2,$$ the form you were asking about.

In turn, this allows us to define all kinds of useful laws involving generalized restoring forces relying linearly—up to a point—on generalized displacements.

One example is $F=kx$ for springs, a customized version of Hooke's Law $\sigma=E\varepsilon$ in which the generalized force is the actual force $F$ and the generalized displacement is the actual displacement $x$. The strain energy is $dU=F\,dx$, which is integrated to obtain $U=\frac{1}{2}kx^2$.

(Equivalently, since energies are derivatives of forces, one could have worked backward from $U\propto\frac{1}{2}x^2$ as justified above to obtain $F=E^\prime\propto x$, the spring law.)

Another example is $V=IZ$, complex Ohm's Law, where the generalized force is the voltage and the generalized displacement is the current (rate of flow of charge $Q$). The impedance $Z$ in an LC circuit is related to $L$ and $C$, and ultimately we find that the total energy is $U=\frac{1}{2}CV^2+\frac{1}{2}LI^2$, which can be transformed into your expression using the relationships between time-dependent voltage and current at the resonant oscillation frequency $\omega_0=\frac{1}{\sqrt{LC}}$.

  • $\begingroup$ So the "resemblance" is only due to our mathematical reasoning and derivations i.e a Taylor series expansion, that you pointed out. And there is no natural physical meaning behind the similarity of the two terms? $\endgroup$ Jul 21, 2021 at 18:35
  • $\begingroup$ The answer to that is subjective, as everyone has different definitions of "natural." All we have to describe Nature is our models. In this case, the concepts of "energy," "potential well," "minimization," and "equilibrium" apply to both spring and circuit oscillation, plus a large number of other oscillatory phenomena (such as the pendulum, for instance). $\endgroup$ Jul 21, 2021 at 19:22

In my opinion, it is because the way we define $k$ i.e. the spring (or equivalent) constant or the force required to displace a unit length.
The force required is $F=kx$, where $x$ is displacement from equilibrium position. Now, energy spent or work done is $\text{d}W = F\text{d}x$. So total work done is $W=\int F\text{d}x = \int_0^x kx\text{d}x$. Therefore, $W=kx^2/2$. That Energy = Force * displacement (from equilibrium) is a universal and hence we find equivalent formula for other energy forms.


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