Estimating the Dipole Radiation Power using Dimensional Analysis

An Oscillating dipole Emitting Radiation
An electric dipole is characterized by the dipole moment $d$ which depends on both the charge $e$ and a the separation $s$ (but not independently on each).
Estimating the power radiated by an oscillating electric dipole is an excellent example for the usage of dimensional analysis.

Let us assume we have a dipole oscillating with frequency $\omega$. The relevant equations to describe the problem are obviously Maxwell's equations. Thus, unlike the electrostatic description of the Hydrogen Atom, the speed of light $c$ will be relevant for the solution. Moreover, instead of an expression that depends on the electron charge $e$, we will seek one which depends on the dipole moment $d$, which is charge times displacement: $\left[d\right] = esu \cdot cm$. The expression will obviously depend on $\omega$ as well. It wont depend on $\hbar$ because the problem is not a quantum mechanical one. Likewise, the expression will also be independent of $G$ because the problems is unrelated to gravity.

Characteristics of an LC circuit using Equipartition

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 average energy in the capacitor is: $$

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