An interesting question about the divergence of a current density

In summary: Are you asking if the current density can be uniform or not uniform in a "continuous loop circuit"? It seems to me that if you are to make that determination, you need to be dealing with a specific circuit. I'm not sure I understand you fully.
  • #1
mertcan
345
6
Hi, maybe as you know ##\nabla. J = -\frac {\partial p} {\partial t}## where J is current density p is charge density.
But also we know current density flux outward the circuit is 0 because current density does not flow out of circuit an this actually volume integral of ##\nabla. J## is zero ( stokes theorem ). NOW here we say that ##\nabla. J## must be zero to make integral 0. But for some infinitesimal volume ##\nabla. J## may be +5, for another infinitesimal volume ##\nabla. J## may be -5 OR for some infinitesimal volume ##\nabla. J## may be +10 for another infinitesimal volume ##\nabla. J## may be -8 and for another infinitesimal volume ##\nabla. J## may be -2. As you see volume integral of ##\nabla. J## is zero in total (10+(-8+(-2)) or +5+(-5)). Could you express to me how the situation that for some infinitesimal volume ##\nabla. J## may be +10 for another infinitesimal volume ##\nabla. J## may be -8 and for another infinitesimal volume ##\nabla. J## may be -2 exists in CONTINUOUS LOOP CİRCUIT (NO CAPACITANCE)?? It seems mathematically valid but I can not imagine the reflect on real world I HAVE NOT COME ACROSS such a (continuous) loop circuit...
 
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  • #3
anorlunda said:
You can extend that down to individual electrons. But in large scale analysis, we make simplifying assumptions. For example, see this PF Insights article. https://www.physicsforums.com/insights/circuit-analysis-assumptions/
I see that maxwell equations at specific point are really accurate, but in order to make great accurate circuit analysis using computational electromagnetism including ONLY maxwell equations is sufficient??
ıf not sufficient what do we do to make real life circuit analysis and what kind of equations are employed??
 
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  • #4
mertcan said:
ıf not sufficient what do we do to make real life circuit analysis and what kind of equations are employed??

Did you read the article linked in post #2? Your question is the topic of the article.

My own profession is analysis of the world's power grids including connected loads. There are no circuits larger than that. The primary tools needed are Ohm's Law and Kirchoff's Laws plus conservation of energy.
 
  • #5
mertcan said:
As you see volume integral of ∇.J∇.J\nabla. J is zero in total (10+(-8+(-2)) or +5+(-5))
Why are you doing a volume integral of ##\nabla \cdot J## ? As far as I know that is not a useful quantity.

As @anorlunda mentioned, the continuity equation leads to Kirchoffs current law.
 
  • #6
Dale said:
Why are you doing a volume integral of ##\nabla \cdot J## ? As far as I know that is not a useful quantity.

As @anorlunda mentioned, the continuity equation leads to Kirchoffs current law.
Just I would like to understand circuit analysis in a deep way I do not want to know shallowly.
 
  • #7
mertcan said:
Just I would like to understand circuit analysis in a deep way I do not want to know shallowly.
Doing random integrals without a firm purpose will not help with that.

Do you have already a firm grasp of Maxwell’s equations?
 
  • #8
mertcan said:
Just I would like to understand circuit analysis in a deep way I do not want to know shallowly.

It sounds like you want to understand conductance more than circuits. Try studying The Drude Model.
https://en.wikipedia.org/wiki/Drude_model
 
  • #9
In the OP, you can use Gauss's law to write ## \int\limits_{V} \nabla \cdot \vec{J} \, d^3 x=\int\limits_{V} \vec{J} \cdot \hat{n} \, dA=-\frac{dQ}{dt} ##. Circuit theory assumes ## \vec{J} ## is uniform everywhere, and without any capacitors you will have ## \int\limits_{V} \vec{J} \cdot \hat{n} \, dA ## will be zero in all cases: what flows into any enclosed volume also flows out of it. (I believe it is Gauss's law that you are needing in the OP, and not Stokes' theorem). ## \\ ## Of course, in the case of capacitors, you can choose a volume for the integral in such a way, with one of the surfaces between the capacitor plates, so that the integral ## \int\limits_{V} \vec{J}\cdot \hat{n} \, dA=-\frac{dQ}{dt} ## gives minus the rate of change of the charge that is collecting on one plate of the capacitor.
 
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  • #10
Charles Link said:
In the OP, you can use Gauss's law to write ## \int\limits_{V} \nabla \cdot \vec{J} \, d^3 x=\int\limits_{V} \vec{J} \cdot \hat{n} \, dA=-\frac{dQ}{dt} ##. Circuit theory assumes ## \vec{J} ## is uniform everywhere, and without any capacitors you will have ## \int\limits_{V} \vec{J} \cdot \hat{n} \, dA ## will be zero in all cases: what flows into any enclosed volume also flows out of it. (I believe it is Gauss's law that you are needing in the OP, and not Stokes' theorem). ## \\ ## Of course, in the case of capacitors, you can choose a volume for the integral in such a way, with one of the surfaces between the capacitor plates, so that the integral ## \int\limits_{V} \vec{J}\cdot \hat{n} \, dA=-\frac{dQ}{dt} ## gives minus the rate of change of the charge that is collecting on one plate of the capacitor.
@Charles Link, you also agree that for some infinitesimal volume ##\nabla. J## may be +10 for another infinitesimal volume ##\nabla. J## may be -8 and for another infinitesimal volume ##\nabla. J## may be -2 in real life in given continuous loop circuit (without capacitors for instance...)?
 
  • #11
mertcan said:
@Charles Link, you also agree that for some infinitesimal volume ##\nabla. J## may be +10 for another infinitesimal volume ##\nabla. J## may be -8 and for another infinitesimal volume ##\nabla. J## may be -2 in real life in given continuous loop circuit (without capacitors for instance...)?
I'm not sure that unless you are referring to quantum fluctuations that you would have that much variation in the current density ## \vec{J} ## to cause ## \nabla \cdot \vec{J} ## to show that kind of variation.
 

Related to An interesting question about the divergence of a current density

1. What is the definition of divergence of current density?

The divergence of a current density is a measure of the rate at which electric charge is flowing out of or into a given point in space. It is a vector field that describes the behavior of electric currents in a given region.

2. How is the divergence of current density calculated?

The divergence of current density is calculated using the vector calculus operation of taking the dot product of the vector field and the del operator (∇). This results in a scalar value representing the net flow of electric charge at a given point in space.

3. What are the units of divergence of current density?

The units of divergence of current density are typically expressed in amperes per meter squared (A/m²). This is because the divergence represents the change in current density over a unit area.

4. What is the significance of the divergence of current density in physics?

The divergence of current density is an important concept in electromagnetism and is used to understand the behavior of electric currents and their effects on the surrounding environment. It is also a key component in Maxwell's equations, which describe the fundamental laws of electromagnetism.

5. How does the divergence of current density relate to the flow of electric charge?

The divergence of current density is directly related to the flow of electric charge. A positive divergence value indicates that electric charge is flowing out of the point in space, while a negative value indicates that charge is flowing into the point. This concept is crucial in understanding the behavior of electric circuits and electromagnetic fields.

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