Pre-Quantum Electrodynamics
Energy in Systems of Point Chargesems.es.efo.e
Work; Pairwise Energy
Consider some distribution of charge which produces an electric field. How much work is needed to carry a small test charge \(q_t\) from one place to another, i.e. from point \({\bf a}\) to point \({\bf b}\) ?
The work done in carrying this charge along some path is the negative of the electrical force in the direction of motion, so
\[ W = -\int_{{\bf a}}^{{\bf b}} {\bf F} \cdot d{\bf l} \]
Let's consider for simplicity a fixed source particle of charge \(q_s\) at position \({\bf r}_s \equiv {\bf 0}\). The work done against electrical forces when moving a unit charge test particle from \({\bf a}\) to \({\bf b}\) is then:
\[ -\int_{\bf a}^{\bf b} {\bf F} \cdot d{\bf l} = -\frac{q_t q_s}{4\pi \varepsilon_0} \int_{\bf a}^{\bf b} \frac{\bf r}{r^3} \cdot d{\bf r} \]
For the path, the 'angular' part is not contributing (see drawings in FLS II 4-3). The integral is thus purely radial,
- Gr (2.18)
As should be clear by now, this result does not depend on the path (if it did, we'd have a perpetuum mobile when going from \({\bf r}_a\) to \({\bf r}_a\) one way, and coming back another).
When thinking about the energy of this pair of charges, we think of starting from an initial configuration where the charges are infinitely distance (which we associate to zero energy), and thus set \({\bf r}_a = \infty\). We can thus write
\[ W = \frac{1}{4\pi \varepsilon_0} \frac{q_t q_s}{|{\bf r}_t - {\bf r}_s|} \]
Example: fission
Generic assembly
By the superposition principle, the energy of a generic assembly of charges \(\{ q_i \}\), \(i = 1, .. n\) (sitting at positions \({\bf r}_i\)) is obtained by the pairwise sum (counting each pair only once) of pairwise energies:
\[ W = \frac{1}{4\pi \varepsilon_0} \sum_{i=1}^n \sum_{j > i}^n \frac{q_i q_j}{|{\bf r}_i - {\bf r}_j|} = \frac{1}{8\pi \varepsilon_0} \sum_{i=1}^n \sum_{j \neq i}^n \frac{q_i q_j}{|{\bf r}_i - {\bf r}_j|} \label{Gr(2.41)} \]
where in the second equality we have symmetrized the sums for convenience.
Crystal lattices
PM3:1.6
Created: 2022-02-07 Mon 08:02