Electric Field from Charge Densities: Gauss's Law

[SpY]]

There are basically three types of charge densities; linear $$\lambda$$ (over a thin ring or rod), surface $$\sigma$$ over a thin flat surface, and over a volume $$\rho$$

The problem is how these densities produce an electric field: applying Coulomb's law for the electrostatic force using appropriate differentials to match the geometry. The first two aren't two bad; but it's the charge in a sphere (as well as out) done in spherical co-ordinates and using Gauss' law in differential form to find the charge density $$\rho$$, and then in integral form to find the total charge.

 

[AJ Bentley]

So what's the question?

 

[SpY]]

It's about electric fields set up by these charge densities..
for example finding the field caused by a charged ring, charged disk or charged sphere

p.s. can't edit topic title typo :/

 

[LawlQuals]

Did you want general advice?

Electric field calculations via Coulomb's law are computed via the following:

$$\vec{E}= \frac{1}{4\pi\varepsilon_0}\int_{q}\frac{dq}{r_{12}^2}\hat{r}_{12}$$

where $$r_{12} = \vec{r}' - \vec{r}$$, and primed coordinates represent locations to the source (charge carrier), and unprimed is the observation point (where you wish to find the field). Also, the quantity $$q$$ is the total charge in the configuration, while $$dq$$ is a differential element of charge.

The choice of which charge density to use is (or will be in time) natural:

$$dq = \lambda d\ell' = \sigma da' = \rho dV'$$

We only integrate over the source coordinates because that is the only place where there is any charge.

You may find it convenient to use each representation above, when you are computing charges residing on: lines, surfaces, and solids respectively.

So, for a sphere, it is useful to use the volume charge density, because it is a solid.

But, I am unable to understand what the question is here. Please advise if I have not answered it properly.

 

[PJC01]

I would like to clarify that the concept of charge density and its relationship to the electric field is a fundamental principle in electromagnetism. Gauss's Law, which relates the electric field to the charge enclosed by a closed surface, is a powerful tool for understanding the behavior of electric charges.

First, it is important to note that the three types of charge densities mentioned, linear, surface, and volume, are all valid ways of describing the distribution of charge. Each type is applicable to different geometries and can be used to calculate the electric field at a certain point.

For linear charge densities, we can use Coulomb's law to calculate the electric field at a point along a thin rod or ring. This is a straightforward application of the law, using appropriate differentials to account for the geometry.

Surface charge densities, on the other hand, require the use of a differential form of Gauss's Law to calculate the electric field. This involves taking the divergence of the electric field and equating it to the surface charge density. This method is also applicable for calculating the electric field at a point on a thin flat surface.

The most complex scenario is when dealing with charge densities in a volume, as in the case of a charged sphere. In this case, we use a combination of Coulomb's law and Gauss's Law in both differential and integral forms. By considering the electric field at a point on the surface of the sphere, we can use the differential form of Gauss's Law to calculate the charge density at that point. Then, by integrating over the entire volume of the sphere, we can find the total charge enclosed.

In summary, the concept of charge density and its relationship to the electric field is a crucial aspect of understanding electromagnetism. By using appropriate equations and methods, we can accurately calculate the electric field from different types of charge densities and gain a deeper understanding of the behavior of electric charges.