# Proof of Gauss's law, starting from Coulomb's Law?

• Dez1
In summary: Also, this case is not hard to show (it just requires basic integration).In summary, The conversation was about deriving Gauss's law from Coulomb's law for a point charge. The steps involved using Coulomb's law to find the electric field, then applying the divergence theorem and the delta function formula to show that Gauss's law is satisfied. It was suggested to first try the simple case of one stationary charge at the origin.
Dez1

## Homework Statement

Provide a proof of Gauss’s law starting from Coulomb’s law for a point charge

## Homework Equations

F=kq1.q2/r^2

(closed)s∫E.n.da=qenc/ε0

## The Attempt at a Solution

My starting point is:

F21=1/(4∏.ε0).(q2.q1)/r21^2.r(unit vector)21

*Note, the 21 terms after F and r should be subscripts

and here's where I've ended up:

E*vector*(r*vector*)=[F*vector*(r*vector*)]/q0= 1/(4∏.ε0).Ʃi(qi)/ri0^2.r(unit vector)i0

Not sure if I'm even on the right track. If anyone could provide the proof it'd be much appreciated.

Coulomb's law for a point charge in the origin is:

$\mathbf{E}\left( \mathbf{r} \right)=\frac{q}{4\pi {{\varepsilon }_{0}}}\frac{{\mathbf{\hat{r}}}}{{{r}^{2}}}$

When you integrate the field on an arbitrary closed surface (that includes the origin):

$\oint\limits_{S}{\mathbf{E}\cdot d\mathbf{S}}=\frac{q}{4\pi {{\varepsilon }_{0}}}\oint\limits_{S}{\frac{1}{{{r}^{2}}}\mathbf{\hat{r}}\cdot d\mathbf{S}}$

because of the inner product $\mathbf{\hat{r}}\cdot d\mathbf{S}$ , the surface element $d\mathbf{S}$ projects onto a spherical surface of radius $r$ , so the surface integral is equivalent to:

$\oint\limits_{S}{\frac{1}{{{r}^{2}}}\mathbf{\hat{r}}\cdot d\mathbf{S}}=\int_{\theta =0}^{\pi }{\int_{\varphi =0}^{2\pi }{\frac{1}{{{r}^{2}}}{{r}^{2}}\sin \theta d\varphi d\theta }}=4\pi$

Alternatively, you could start from the delta function formula:

$\nabla \cdot \left( \frac{{\mathbf{\hat{r}}}}{4\pi {{r}^{2}}} \right)=\delta \left( \mathbf{r} \right)$

to evaluate the divergence of the electric field:

$\nabla \cdot \mathbf{E}\left( \mathbf{r} \right)=\frac{q}{{{\varepsilon }_{0}}}\delta \left( \mathbf{r} \right)$

Then, applying the divergence theorem in the volume integral of $\nabla \cdot \mathbf{E}\left( \mathbf{r} \right)$ , you get:

$\oint\limits_{S=\partial V}{\mathbf{E}\cdot d\mathbf{S}}=\int\limits_{V}{\nabla \cdot \mathbf{E}dV}=\int\limits_{V}{\frac{q}{{{\varepsilon }_{0}}}\delta \left( \mathbf{r} \right)dV}=\frac{q}{{{\varepsilon }_{0}}}$

Easy way is to assume Gauss' law is right, and derive Coulomb's law from it.

Gauss' law is more general than Coulomb's law, because in Coulomb's law, we assume stationary charges. So, for the title of this thread, I think it should be: "Show that if Coulomb's law holds, then so does Gauss' law". Anyway, sorry to be nitpicky.

Dez1, since this is the homework section, we are meant to offer help, but not to give you the answer. It will be better practice if you try to have a good attempt at the problem yourself. Your working is looking good so far. Your final line of working (if I read it right) is:
$$\vec{E}= \frac{1}{4 \pi \epsilon_0} \sum \frac{q_i}{{r_{i0}}^2} \hat{r_{i0}}$$
What you have done so far is correct. I think first you should try the simple case of when there is just one charge. Also, since the system has translational symmetry, you can (without loss of generality), say that the position of the single charge is at the origin. So this will simplify your equation nicely.

Once you've done this, you have the electric field (from Coulomb's law). So now, you need to use that in Gauss' law, and show that Gauss' law is satisfied. Or, equivalently, you can use it in the integral form of Gauss' law, and show that it is satisfied for any arbitrary region.

Hint: the proof is mostly just maths. another hint: you will need to make use of the properties of the Dirac-delta function.

Edit: It is not difficult to extend to the case when there are several stationary point charges. But (at least for me, anyway), it is best to think about the simple case first.

Another Edit: Also, if you have not learned about Dirac-delta functions yet, you can still show that the integral form of Gauss' law holds for a spherical Gaussian surface centred on the point charge. But of course, this is not 'proving' Gauss' law fully.

Last edited:

## 1. What is Gauss's law and how does it relate to Coulomb's law?

Gauss's law is a fundamental law in electromagnetism that relates the electric flux through a closed surface to the charge enclosed within that surface. This law is based on Coulomb's law, which describes the force between two point charges. Gauss's law is a more general form of Coulomb's law and can be used to solve for the electric field for more complex charge distributions.

## 2. How is Gauss's law mathematically expressed?

Gauss's law is mathematically expressed as ∮E⃗·dA = Qenc0, where ∮E⃗·dA represents the electric flux through a closed surface, Qenc is the total charge enclosed within that surface, and ε0 is the permittivity of free space.

## 3. What are the assumptions made in Gauss's law?

Gauss's law assumes that the electric field is symmetrical around the charge distribution and that the charge is uniformly distributed. It also assumes that the closed surface used to calculate the electric flux is a regular shape, such as a sphere or a cube.

## 4. Can Gauss's law be used to solve for the electric field in all situations?

No, Gauss's law can only be used to solve for the electric field in situations where the charge distribution and the closed surface satisfy the assumptions of the law. In more complex situations, other methods such as integration of Coulomb's law must be used to calculate the electric field.

## 5. What are some real-world applications of Gauss's law?

Gauss's law has many practical applications, such as in the design of capacitors, in the calculation of the electric field of a point charge, and in the analysis of electric fields in conducting materials. It is also used in the study of planetary and celestial bodies to understand their electric fields and how they interact with other objects in space.

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