Helmholtz's theorem and charge density

In summary, Helmholtz's theorem states that if electric charge density goes to zero as r goes to infinity faster than 1/r^2, I'm able to construct an electrostatic potential function using the usual integral over the source, yet I don't understand how this applies to a chunk of charge in some region of space, regarding how fast its charge density does to zero.
  • #1
Ahmed1029
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According to Helmholtz’s theorem, if electric charge density goes to to zero as r goes to infinity faster than 1/r^2 I'm able to construct an electrostatic potential function using the usual integral over the source, yet I don't understand how this applies to a chunk of charge in some region of space, regarding how fast its charge density does to zero. For instance, suppose I only have a small sphere of charge, how fast does does charge density go to zero as r goes to infinity?
 
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  • #2
If you have a charge density different from 0 only within a finite region in space (mathematically speaking, ##\rho## has "compact support"), the assumptions of the theorem are fulfilled, because it's strictly 0 for ##r=\vec{x}>R## for some radius, ##R##. Then the solution of the Poisson equation (in SI units),
$$\Delta \Phi(\vec{x})=-\frac{1}{\epsilon_0} \rho(\vec{x})$$
is given by "summing up Coulomb potentials", i.e.,
$$\Phi(\vec{x})=\int_{B_R} \mathrm{d}^3 x' \frac{\rho(\vec{x}')}{4 \pi \epsilon_0 |\vec{x}-\vec{x}'|},$$
where ##B_R## is the solid sphere ("ball") of radius ##R## around the origin, which by definition contains all the charge there is.

Helmholtz's theorem in this form is for sure also valid under the conditions you quote, because if ##\rho(\vec{x}) = \mathcal{O}(r^{-\alpha})## for ##|\vec{x}|\rightarrow \infty## and ##\alpha>2##, then
$$\mathrm{d}^3 x' \frac{\rho(\vec{x}')}{|\vec{x}-\vec{x}'|} = \mathrm{d} r' \mathrm{d\vartheta'} \mathrm{d} \varphi' \sin \vartheta' \mathcal{O}(r^{\prime -\alpha+1}),$$
i.e., the integrand falls faster with ##r'## than ##1/r##, and thus the integral over ##r'## converges for ##r' \rightarrow \infty##. The integrals over the angles is over finite regions (##\vartheta \in (0,\pi)##, ##\varphi \in (0,2 \pi)##) and thus don't diverge.

The theorem even works if ##\rho=\mathcal{O}(1/r^\beta)## with ##\beta>1##, because the potential is defined only up to an additive constant anyway. In this case you just subtract a clever constant, defining
$$\Phi(\vec{x})=\int_{\mathbb{R}^3} \mathrm{d}^3 x' \frac{\rho(\vec{x}')}{4 \pi \epsilon_0} \left (\frac{1}{|\vec{x}-\vec{x}|}-\frac{1}{|\vec{x}_0-\vec{x}'|} \right).$$
The expression in the parentheses goes like ##\mathcal{O}(r^{\prime 2})## for ##r' \rightarrow \infty##, and thus the integral converges even under the more general condition on ##\rho##.
 
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  • #3
got it thanks
 
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Related to Helmholtz's theorem and charge density

1. What is Helmholtz's theorem?

Helmholtz's theorem is a fundamental principle in vector calculus that states any vector field can be decomposed into two components: a solenoidal component (divergence-free) and an irrotational component (curl-free).

2. How is Helmholtz's theorem related to charge density?

In electrostatics, Helmholtz's theorem is used to express the electric field generated by a charge distribution in terms of its charge density. This allows for a simplified calculation of the electric field in a given region.

3. What is charge density?

Charge density is a measure of the amount of electric charge per unit volume in a given region. It is commonly denoted by the symbol ρ and is expressed in units of coulombs per cubic meter (C/m³).

4. How is charge density calculated?

Charge density can be calculated by dividing the total charge in a given region by the volume of that region. It can also be calculated by taking the limit of the charge enclosed by an infinitesimally small volume as the volume approaches zero.

5. What are some applications of Helmholtz's theorem and charge density?

Helmholtz's theorem and charge density are used in many areas of physics and engineering, such as electrostatics, electromagnetism, fluid dynamics, and acoustics. They are also important in the study of potential theory and the solution of differential equations.

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