Differential Form of Gauss's Law

In summary, the differential form of Maxwell's equations is just the same as the integral form, but with the milli-smidgen of volume of the little cube being arbitrarily small.
  • #1
Feldoh
1,342
3
Could someone try and explain with the differential form means? I've only taken p to calculus 2 so I'm not really sure what divergence in the sense of this equation means. Also what is the difference in the two. I mean the integral form looks at an electric field and charge over a region, so what does the differential form represent?
 
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  • #2
Feldoh said:
Could someone try and explain with the differential form means? I've only taken p to calculus 2 so I'm not really sure what divergence in the sense of this equation means. Also what is the difference in the two. I mean the integral form looks at an electric field and charge over a region, so what does the differential form represent?

Always in physics we use to write equation in differential form because they explain how it is the behavior of a system locally...
the div operator acting on a vector field suggest us if the field is divergenceless (solenoidal) or not. Physically this mean that it counts the source or sink of the field. the abjective soleinoidal come from the magnetic field B, in fact we know that divB=0!

To obtain the integral equation we just use the divergence theorem, and we obtain the relation with the flux of the field trough a surface...

i think you can find many things on wiki or somewhere else that explain things better than me.

regards
marco
 
  • #3
The differential forms of Maxwell's equations, like Gauss's Law help tell us how a field behaves at a point, which the integral forms cannot tell us about. Other than this one difference, they describe the same physical phenomena. Whether you use one form or another depends on how useful that form is to the problem your working on.

The divergence is just a vector derivative:

[tex]\nabla\cdot\vec{v}= \frac{\partial v_x}{\partial x}+\frac{\partial v_y}{\partial y}+\frac{\partial v_z}{\partial z}[/tex] In fact, just as there are two ways to multiply vectors (dot and cross products), there are two ways to differentiate them. You can take a vector's divergence, or it's curl. Both are derivatives, but they tell you different things. The divergence tells you how much the vector field diverges from a point, i.e. the electric field from a positive point charge had a high divergence. (It always points away from the source of the field. In other words, it diverges from that point.) On the other hand, the magnetic field of an infinite current-carrying wire, which loops around the wire has zero divergence (the field lines are always the same distance from the wire), but they have a high curl (the field lines "curl" back on themselves.)
 
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  • #4
The differential form and the integral form are the same thing. Once you learn the divergence theorem it will be plain as daylight.

Edit: Of course the differential form is much more pretty, and describes what is happening as a point, as others already mentioned.
 
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  • #5
You find the differential form of Maxwell's equatons prettier? I find the integral forms easier to understand and more enlightening.
 
  • #6
Defennnder said:
You find the differential form of Maxwell's equatons prettier? I find the integral forms easier to understand and more enlightening.

think of the differential form as exactly the same as the integral form, but with the milli-smidgen of volume of the little cube being arbitrarily small. it's the same thing.
 

Related to Differential Form of Gauss's Law

What is the Differential Form of Gauss's Law?

The Differential Form of Gauss's Law is an equation used in electromagnetism to describe the relationship between electric charges and the electric field. It is a fundamental law of physics that states that the electric flux through a closed surface is equal to the total charge enclosed by that surface.

How is the Differential Form of Gauss's Law different from the Integral Form?

The Differential Form of Gauss's Law is an equation that describes the relationship between the electric field and the charge distribution at a single point in space, while the Integral Form describes the relationship between the electric field and the charge distribution over a closed surface.

What is the mathematical representation of the Differential Form of Gauss's Law?

The mathematical representation of the Differential Form of Gauss's Law is ∇ · E = ρ / ε0, where ∇ is the divergence operator, E is the electric field, ρ is the charge density, and ε0 is the permittivity of free space.

What is the meaning of the terms in the Differential Form of Gauss's Law equation?

The term ∇ · E represents the divergence of the electric field, which is a measure of how much the electric field is spreading out or converging at a particular point. The term ρ represents the charge density, which is the amount of charge per unit volume, and ε0 is the permittivity of free space, which is a constant value representing the ability of a material to store electric charge.

How is the Differential Form of Gauss's Law used in practical applications?

The Differential Form of Gauss's Law is essential in understanding and predicting the behavior of electric fields in various systems, such as circuits, antennas, and electric motors. It is also used in the study of electromagnetism and in the design of electronic devices.

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