[Physics] Generalized Coulomb’s Law

coulombs-lawelectrostatics

This question is about the generalization of Coulomb's law to continuous bodies of charge. The basic statement of Coulomb's Law involves two discrete charges $q_1$ an $q_2$:

$$\vec{F}_i = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r_{12}} \hat{r}_i $$

Here $i$ represents the charge on which the force is exerted, and $\hat{r}_i$ represents the unit displacement vector between the other charge and the charge $i$.

Many treatment of electrostatics extend this law to the case that one charge is not discrete, but rather a continuous body. The force on the discrete charge $Q$ is then:

$$\vec{F} = \frac{Q}{4 \pi \epsilon_0} \int \frac{dq}{r^2} \hat{r} $$

Here $dq$ is the infinitesimal charge element of the continuous body, while $r$ and $\hat{r}$ represent the distance and displacement vectors between $dq$ and $Q$.

Continuing this way, we could probably propose an expression for force between two continuous bodies of charge, like so:

$$\vec{F} = \frac{1}{4 \pi \epsilon_0} \int \int \frac{dq_1 dq_2}{r^2} \hat{r}$$

However, I have not really seen this expression in the literature/treatments of electrostatics. Does anyone know why this is the case? Is the expression not useful, or are there no applications demanding the above expression?

Best Answer

I am not quite sure which literature are you looking for, but it should be written in standard textbook. Anyway, you are right that it is not that useful except a formulation and exercise.

Usually, it is written in the following form. Given a charge distribution $\rho_{1}(\vec{\mathbf{r}_{1}})$ for object with volume $V_1$ and $\rho_{2}(\vec{\mathbf{r}_{2}})$ for object with volume $V_2$, we have the electric field: $$\vec{\mathbf{E}}(\vec{\mathbf{r}_{2}})=\frac{1}{4\pi\epsilon_{0}}\int_{V_{1}}\frac{\rho_{1}(\vec{\mathbf{r}_{1}})}{|\vec{\mathbf{r}_{2}}-\vec{\mathbf{r}_{1}}|^{3}}(\vec{\mathbf{r}_{2}}-\vec{\mathbf{r}_{1}})d\vec{\mathbf{r}_{1}} \tag{1}$$ and force $$\vec{\mathbf{F}}=\frac{1}{4\pi\epsilon_{0}}\int_{V_{2}}\int_{V_{1}}\frac{\rho_{1}(\vec{\mathbf{r}_{1}})\rho_{2}(\vec{\mathbf{r}_{2}})}{|\vec{\mathbf{r}_{2}}-\vec{\mathbf{r}_{1}}|^{3}}(\vec{\mathbf{r}_{2}}-\vec{\mathbf{r}_{1}})d\vec{\mathbf{r}_{1}}d\vec{\mathbf{r}_{2}} \tag{2}$$

Because of practicality, it is not used that often:

  1. It is hard to apply on non-fixed charge distribution. For example, for conductor, the charge itself would change as two object approaching. This internal dynamics is not captured in Eq (2).
  2. When the objects are far away, it is just like a point charge. Why do you want to do the cumbersome integration?
  3. There are better tools to handle this situation: Multipole expansion. As two objects become closer and closer, you include the monopole first (i.e. a point charge), then dipolar, then quadrupole... This expansion is very systematical and have good physical meaning.
  4. People care more about electric field rather than force, as it is more fundamental, so the Eq (1) is more emphasized than Eq (2).
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