An LC circuit is one which has a capacitor and an inductor connected to each other. It exhibits oscillations just like a mass on a spring (a harmonic oscillator). In fact, the analogy is quite accurate with the capacitor playing the role of the spring and the inductor playing the role of the mass inertia.
Just like any harmonic oscillator, we can use equipartition to estimate the energy and frequency of oscillations using equipartition.
The standard way to obtain the size of the hydrogen atom, also known as Bohr's radius, is to solve Schrödinger equation for the hydrogen atom. This is a somewhat detailed calculation requiring the usage of generalized Laguerre polynomials and spherical harmonics. We can however bypass it, if we are only interested in an estimate of the hydrogen atom.
One method which we don't follow here, is to estimate the size of the atom using dimensional analysis. Instead, we shall do so using the principle of equipartition.
Ice covering Grenadier Lake, High Park Toronto. The ice was 8 cm thick, enough to stand on. The small patch in the ice is a dead fish, frozen into the ice. How cold should it be and for how long to have ice thick enough to stand on?
Many physical systems have a tendency to equilibrate the energy between different subcomponents. Sometimes it is exact, and sometimes not. For example, in an acoustic wave, the wave's energy is on average half kinetic (motion of the gas) and half internal (pressure). In the interstellar medium, there is roughly the same energy in the different components, such as internal energy, turbulent energy, magnetic field and energy of the cosmic rays. Stars are no different. In the sun, there is roughly the same binding energy (which is negative) as there is thermal energy. This can also be shown using the virial theorem. In white dwarfs, the thermal energy is unimportant, instead, there the degenergy energy of the electrons is comparable to the binding energy. We can use this tendency for equipartition to estimate different stellar parameters.
