Confusion about Gauss' law differential form

In summary, the Gauss' law in differential form states that the divergence of the electric field is equal to the charge density divided by the permittivity of free space. However, at locations where charges are present, the charge density is represented by a Dirac delta function. When dealing with condensed matter, the sum of single-charge delta distributions can be coarse grained to a continuous charge distribution.
  • #1
constfang
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http://einstein1.byu.edu/~masong/emsite/S1Q50/EQMakerSL1.gif

Hi guys, I have a little confusion about the Gauss' law in differential form over here, obviously, many textbook wrote it in the above form, but actually, the only place at which divE is not zero is at the locations where the charges are present. so I read over here: http://farside.ph.utexas.edu/teaching/em/lectures/node30.html#e343 (equation 208) that there should be a Dirac delta function over there, is it correct? then why does people keep omitting that Dirac delta function and just quoted it as above, where no information regarding the position of the point where we're calculating the divergence is given? Thank you.
 
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  • #2
The fundamental law, using SI units, is indeed

[tex]\vec{\nabla} \cdot \vec{E}=\frac{\rho}{\epsilon_0}.[/tex]

Here, [itex]\rho[/itex] is the charge density of the charged matter, producing the electromagnetic field.

If the matter consists of a single charge at rest, this density is a [itex]\delta[/itex] distribution

[tex]\rho_{\text{single charge}}(\vec{x})=q \delta^{(3)}(\vec{x}-\vec{x}_0).[/tex]

If you have condensed matter, you can coarse grain the sum of the single-charge [itex]\delta[itex] distributions to a continuous charge distribution.
 
  • #3
Oh I got it now, thank you very much.
 

1. What is Gauss' law in its differential form?

Gauss' law in its differential form is one of the four Maxwell's equations that describes the relationship between electric fields and electric charges. It states that the divergence of the electric field at a point is equal to the charge density at that point divided by the permittivity of free space.

2. How is Gauss' law in its differential form different from the integral form?

The differential form of Gauss' law is a local equation, meaning it applies to a specific point in space. It relates the electric field and charge at a single point. The integral form, on the other hand, is a global equation, meaning it applies to a larger region in space and relates the total electric flux through that region to the enclosed charge.

3. What is the significance of Gauss' law in its differential form?

Gauss' law in its differential form is important because it allows us to calculate the electric field at a specific point in space, given the charge distribution around it. This is useful in solving many real-world problems, such as calculating the electric field inside a charged capacitor or near a charged conducting surface.

4. Can Gauss' law in its differential form be applied to all types of charge distributions?

Yes, Gauss' law in its differential form can be applied to any type of charge distribution, including point charges, continuous charge distributions, and even non-uniform charge distributions. However, the shape of the charge distribution may affect the complexity of the calculations needed to apply the law.

5. How is Gauss' law in its differential form derived?

Gauss' law in its differential form is derived from the integral form of Gauss' law using the divergence theorem. The divergence theorem states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of that field over the region enclosed by the surface. By applying this theorem to the integral form of Gauss' law, we can obtain the differential form.

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