Electric Field for Non-Symmetric Charge Distributions

Question

In real-world situations where charge distributions are asymmetric and continuous, how is the electric field calculated? Gauss's Law in integral form can't be practically applied for asymmetric distributions, and although I've heard it alluded to that the differential form can be used I lack the mathematical skill to understand how (i.e whether the PDE $\nabla \cdot {\bf E}=\frac{\rho}{\epsilon_0}$ can be numerically integrated to find ${\bf E}$). My only other thoughts are using Coulomb's Law or that there's some other method I haven't learned about.

Alternatively, if this has been asked before I'd appreciate a link to the question.

Answer 1

I am guessing from how you phrase the question that you have taken an introductory university-level course in electricity and magnetism; and so you have perhaps seen Gauss's Law and Coulomb's Law, but not much else in the way of problem-solving techniques. If this is not the case, let me know in the comments and I'll modify my answer accordingly.

As a general rule, it is not possible to directly use Gauss's Law to solve for the electric field of an arbitrary charge distribution. The only cases in which it's possible are those with a high degree of symmetry. In such instances, you're frequently stuck with Coulomb's Law.¹

For an asymmetric, continuous charge distribution, the tactic is to:

1. View the continuous charge distribution as a bunch of little chunks with small volumes $\Delta V$.
2. Figure out the charge $\Delta q$ inside each little chunk by the relationship $\Delta q \approx \rho \Delta V$.
3. Figure out the electric field $\Delta \vec{E}$ due to each little chunk at the point $P$ where we want to find the field, using Coulomb's Law.
4. Add up the fields from all the chunks to get the total field $\vec{E} = \sum_\text{chunks} \Delta \vec{E}$.
5. Take the limit as these "chunks" go to infinitesimally small size. In this limit, the sum becomes a multiple integral over the volume occupied by the charge distribution.

The process is tedious but straightforward. The problem is that turning the formal "sum" from step 4 into an integral in step 5 can be rather complicated; and even if you can write down such an integral, it often happens that you can't actually perform the integral to get a result in terms of elementary functions. Even for a situation as simple as the field at a general point in the plane of a ring of charge, you run into things like elliptic integrals; and encountering them in a problem usually requires "a very specific flavour of giving up" on solving your problem.

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¹ There are various more advanced techniques that sometimes be applied, which you'll learn about in upper-level E&M courses. As it happens, one of these techniques (the method of Green's functions) is secretly equivalent to Coulomb's Law in the case of a charge distribution for which the electric potential goes to zero at infinity. It's all related!