An interesting question about the divergence of a current density

[mertcan]

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...

 

[anorlunda]

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/

 

[mertcan]

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??

 

[anorlunda]

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.

 

[Dale]

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.

 

[mertcan]

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.

 

[Dale]

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?

 

[anorlunda]

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

 

[Charles Link]

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.

 

[mertcan]

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...)?

 

[Charles Link]

@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.