Question on the Gauss' Law's Qenc

[IsaacOh]

I understand Gauss' Law is: Phi = integral(E)dA = Qenc/epsilon0

I never understood how you determine that Qenc. I know it varies from shape to shape. Can someone help me?

 

[Born2bwire]

It's simply the total charge enclosed by the Gaussian surface that you chose to integrate over on the left-hand side. If we assume that we are dealing with discrete charges, then it would be summation of the enclosed charges. Otherwise, more generally, it is the integration of the enclosed charge density. As such,

$$\int_\Gamma \mathbf{E} \cdot d\mathbf{S} = \frac{1}{\epsilon} \int_\Gamma \rho(\mathbf{r}) dS$$

 

[tiny-tim]

Welcome to PF!

Hi IsaacOh! Welcome to PF!

I never understood how you determine that Qenc. I know it varies from shape to shape.

Q_enc is simply the charge enclosed by whatever shape the ∫ is over

do you mean, what shape should you choose?

whatever is most convenient (ie, makes the maths simplest, eg because it cuts all the field lines at 0° or 90°)​

(do a google image search for Gauss' law for some pictures)

 

[IsaacOh]

Thanks Born2bwire and Tiny-Tim for replying! My question was how do you determine Qenc and Bor2bwire, I believe, answered that.

 

[vanhees71]

Well, there is a bit of confusion in this answer. First one should mention that this kind of reasoning boils down to the integral theorems of classical vector analysis or, in a more modern way in terms of alternating differential forms, to the general Stokes theorem.

Let's put it in the classical way of 3D Euclidean vector analysis since this is more intuitive and that's what's needed in E&M intro lectures. Gauß's Law connects the volume integral over the divergence of the vector field with the integral of this vector field over the surface integral along the boundary of the volume, i.e.,

$$\int_V \mathrm{d}^3 x \vec{\nabla} \cdot \vec{V} = \int_{\partial V} \mathrm{d}^2 \vec{F} \cdot \vec{V}.$$

Here, the surface-element normal vectors have to be oriented such that they always point out of the volume you integrate over. This is a mathematical theorem valid for any sufficiently well-behaved vector fields, volumes and boundaries.

One application of this mathematical theorem in E&M is to use Gauß's Law of electrodynamics, which is one of the fundamental laws of electromagnetism, i.e., one of Maxwell's equations:

$$\vec{\nabla} \cdot \vec{D}=\rho.$$

Here $\vec{D}$ is the electric flux density and $\rho$ the charge density. Using Gauß's theorem by integrating over a volume clearly gives the integral form of this law:

$$\int_{V} \mathrm{d}^3 x \rho=\int_{\partial V} \mathrm{d}^2 \vec{F} \cdot \vec{D}.$$

On the right-hand side you have, by definition of charge density, the charge enclosed in the volume, $V$, and on the right-hand side the electric flux through the boundary of this same volume. Of course, again you have to orient the suface-normal vectors out of this volume, i.e., the relative orientation of the boundary to the volume must be positive.