Maxwell’s equations with varying charge but constant current

[mertcan]

Hi, initially I would like to share this link related to derivation of maxwell equation : http://www.physicspages.com/tag/amperes-law/

My question is : in this link, it states interesting situation in electrodynamics is one where the current density

is constant in time, but the charge density

isn’t. I do not understand this sentence, because if current density is constant then current should be constant which means steady state current exist, therefore divergence of J should be 0 which means charge density is constant in time but link does not say so (1 ST QUESTİON)

, also as a example of it's statement it gives the example of charging capacitor, but in this situation current varies with time which means not constant current and current density. (2 ND QUESTİON)

Besides, if you look at deeply in the link biot savart law is used (applicable to steady currents) not jefimenko equation and after the curl is taken over biot savart, it says divergence of J is not 0 (equal to charge density change over time) , I mean although biot savart rule exist when steady state currents exist, (for the expansion of curl of biot savart) link says divergence of J is not 0. HOW ARE THESE POSSIBLE??

 

[phyzguy]

Why couldn't you charge a capacitor with a constant current? If you have a capacitor C and you flow a constant current I onto it, this is a physically realizable situation. Of course, it can't continue forever, because eventually the capacitor will blow up, but it could continue for a period of time, even a very long time if I is small enough.

 

[mertcan]

Why couldn't you charge a capacitor with a constant current? If you have a capacitor C and you flow a constant current I onto it, this is a physically realizable situation. Of course, it can't continue forever, because eventually the capacitor will blow up, but it could continue for a period of time, even a very long time if I is small enough.

I do not ask what you meant in your respond, I ask different things related to link...

 

[phyzguy]

I do not ask what you meant in your respond, I ask different things related to link...

I was responding to your (2ND QUESTION), where you stated that in the case of charging a capacitor, "...but in this situation current varies with time which means not constant current and current density." I was stating that this is not necessarily the case; it is possible to charge a capacitor with a constant current. This seems to be the type of situation that is being analyzed in the link you posted.

 

[mertcan]

I was responding to your (2ND QUESTION), where you stated that in the case of charging a capacitor, "...but in this situation current varies with time which means not constant current and current density." I was stating that this is not necessarily the case; it is possible to charge a capacitor with a constant current. This seems to be the type of situation that is being analyzed in the link you posted.

Ok I agree with you but if you look at the link biot savart rule is applied which means that steady current exist (otherwise in time dependent current density jefimenko's equations should be applied for magnetic field), but when curl of magnetic field towards biot savart is applied there are terms including divergence of current density and link says divergence of current density is not 0 is equal to charge density change in time but at the beginning we say that current is steady so there is a ambivilance so I do not understand this situation in link?

Also How is it possible that constant current density/current exist with time varied charge density ? Towards the continuity equation if divergence of current density is zero(steady current) then time varied charge density should e zero ?

 

[phyzguy]

Also How is it possible that constant current density/current exist with time varied charge density ? Towards the continuity equation if divergence of current density is zero(steady current) then time varied charge density should e zero ?

I'm not sure about your first question, but I can answer this one. Steady current does not mean that the divergence of current density is zero. The divergence of the current density is a function of the spatial derivatives of J, not the time derivative of J. In the case of a charging capacitor, J is flowing onto the capacitor plate, but J between the plates is zero. So look at the Gaussian box in the attached drawing. J is flowing into the box, but none is flowing out. So Div J is clearly not zero.

 

[mertcan]

Hi, I would like to express that I have not received any response to my last question (post 7) for a long time. I do not understand this situation. Is there a problem or incomprehensible thing in my question? I really want you to be sure that I have tried my last question to be very explicit. In short let me rewrite my question here
I also would like express that I shared an attachment ( a part of the link I shared at the beginning of my posts, for recall link is: http://www.physicspages.com/tag/amperes-law/ ) In attachment you see 3 boxes, link says that term in the red box vanishes, link also gives the reason which is the bottom box. My QUESTİON is why term in red box vanishes ?
I am asking because I do not understand what the bottom box include ( I mean I do not understand the reason that link shared in the bottom box)

 

[phyzguy]

Hi, I would like to express that I have not received any response to my last question (post 7) for a long time. I do not understand this situation. Is there a problem or incomprehensible thing in my question? I really want you to be sure that I have tried my last question to be very explicit. In short let me rewrite my question here
I also would like express that I shared an attachment ( a part of the link I shared at the beginning of my posts, for recall link is: http://www.physicspages.com/tag/amperes-law/ ) In attachment you see 3 boxes, link says that term in the red box vanishes, link also gives the reason which is the bottom box. My QUESTİON is why term in red box vanishes ?
I am asking because I do not understand what the bottom box include ( I mean I do not understand the reason that link shared in the bottom box)

I think you are misunderstanding. They are not saying that the term in the red box vanishes. They are saying that the integral of the term in the red box over all space vanishes. The reason is that $\int \nabla \cdot F dV$ over a volume is equal to $\int F dS$ over the surface, by Stoke's theorem. In the problem they are analyzing, the surface goes to infinity and the currents J vanish at infinity, so the surface integral is zero. Therefore the volume integral is zero. This is a standard technique. I think this is the answer to your question. Does this make sense?

 

[mertcan]

I think you are misunderstanding. They are not saying that the term in the red box vanishes. They are saying that the integral of the term in the red box over all space vanishes. The reason is that $\int \nabla \cdot F dV$ over a volume is equal to $\int F dS$ over the surface, by Stoke's theorem. In the problem they are analyzing, the surface goes to infinity and the currents J vanish at infinity, so the surface integral is zero. Therefore the volume integral is zero. This is a standard technique. I think this is the answer to your question. Does this make sense?

Ok I understand your response, but I also wonder that do we have to stretch our surface to infinity to make the red box vanish? For instance, if we take our surface as circuit surface instead of all space, can red box be still zero? Because I consider that if we take volume integral of divergence of current density (also means surface integral of current density over circuit surface), then result is zero, because divergence of current density means charge density change in a given time and total charge do not change in circuit as a result of charge conservation law ( inflow of charge in circuit equals outflow of charges in circuit) ?

 

[phyzguy]

I think you are right that we only have to take a surface which is large enough so that the currents on the surface are 0. Then the surface integral of J over that surface vanishes, so the volume integral of Div J over that volume vanishes.

 

[mertcan]

I think you are right that we only have to take a surface which is large enough so that the currents on the surface are 0. Then the surface integral of J over that surface vanishes, so the volume integral of Div J over that volume vanishes.

Besides, I would like to put into words that I have contended for finding some examples where current is not same along the continuous loop circuit ( for instance think about only 1 resistor and 1 voltage included, NOT CAPACITOR INCLUDED by the way ). I mean different points along the 1 resistor and 1 voltage included circuit have DIFFERENT divergence of current density. How can it be possible in continuous loop without capacitor? ? Or it can be possible ? Or when is it possible? ?

 

[mertcan]

Hi, I would like to express that I have not received any response to my last question (post 11) for a long time. I admit that it is weird question and I hope it is understandable for you if it is not PLEASE tell me and I will try to make the question comprehensible for you. In short please repeat my question :
I would like to put into words that I have contended for finding some examples where current is not same along the continuous loop circuit ( for instance think about only 1 resistor and 1 voltage included, NOT CAPACITOR INCLUDED by the way ). I mean different points along the 1 resistor and 1 voltage included circuit have DIFFERENT divergence of current density. How can it be possible in continuous loop without capacitor(continuous loop)? ? Or it can be possible ? Or when is it possible? ?

 

[phyzguy]

Hi, I would like to express that I have not received any response to my last question (post 11) for a long time. I admit that it is weird question and I hope it is understandable for you if it is not PLEASE tell me and I will try to make the question comprehensible for you. In short please repeat my question :
I would like to put into words that I have contended for finding some examples where current is not same along the continuous loop circuit ( for instance think about only 1 resistor and 1 voltage included, NOT CAPACITOR INCLUDED by the way ). I mean different points along the 1 resistor and 1 voltage included circuit have DIFFERENT divergence of current density. How can it be possible in continuous loop without capacitor(continuous loop)? ? Or it can be possible ? Or when is it possible? ?

I don't really understand your question. In a continuous loop circuit, the current is the same everywhere in the circuit. This is Kirchoff's current law, which states:

The principle of conservation of electric charge implies that: At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node.

In a simple loop, there are no nodes, so the current is constant everywhere.

 

[mertcan]

I don't really understand your question. In a continuous loop circuit, the current is the same everywhere in the circuit. This is Kirchoff's current law, which states:

The principle of conservation of electric charge implies that: At any node (junction) in an electrical circuit, the sum of currents flowing into that node is equal to the sum of currents flowing out of that node.

In a simple loop, there are no nodes, so the current is constant everywhere.

Ok please let me simplify my question : is there a situation in which charge conservation is not ensured ? Or kirchhoff law is not applied? ? Or do we have to always apply kirchhoff current rule in circuits ? Or is there any kind of situation or exception where kirchhoff current law is not consistent in some circuits?

 

[phyzguy]

Just a comment - one question mark is enough.

The simple answer is that charge is always conserved - no exceptions. Kirchoff's law will not always apply in the case where there is charge storage, such as the capacitor example you mentioned earlier, or a battery which is charging or discharging. From a circuit standpoint however, Kirchoff's law applies even in the case of a charging capacitor, since current flows into the capacitor, and the same current flows out of the capacitor. However, if you look closely, you will see that current flows into one plate and out of the other plate, but there is no current between the plates. So if you look closely inside the capacitor, Kirchoff's law is violated, but if you consider the capacitor as a circuit element, Kirchoff's law holds. Does this make sense?

 

[Lord Jestocost]

However, if you look closely, you will see that current flows into one plate and out of the other plate, but there is no current between the plates. So if you look closely inside the capacitor, Kirchoff's law is violated...

I am not sure whether one can say it in such a way. As long as a capacitor is charging or discharging, there is a time varying electric field between the capacitor plates. Thus, we have a displacement current which is the result of the time varying electric field. The total current density at every point in the circuit is thus always the sum of the conduction current density (drift of charged carriers) and the displacement current density. The divergence of the total current density is always zero.

 

[phyzguy]

I am not sure whether one can say it in such a way. As long as a capacitor is charging or discharging, there is a time varying electric field between the capacitor plates. Thus, we have a displacement current which is the result of the time varying electric field. The total current density at every point in the circuit is thus always the sum of the conduction current density (drift of charged carriers) and the displacement current density. The divergence of the total current density is always zero.

I agree. If you include the displacement current, then the current is constant even between the capacitor plates.

 

[mertcan]

Thanks for your responses and time

 

[mertcan]

Hi, Initially I would like express that curl of magnetic field is dependent on current density (let's think non changed current at first), but also maxwell shows that curl of B is dependent on change in electric field multiplied bu dialectic constant. My question is : If we have continuous loop/circuit (without capacitor for instance) and changed current (AC current for instance), then our curl of magnetic field is dependent on both current density ( conductivity of wire * electric field in wire) and change of electric field multiplied by free space permittivity constant of wire? Also I saw in some sites that change of electric field in wire may be negligible...

 

[mertcan]

besides my last question I also would like to express that I tried to obtain curl of time dependent magnetic field (AC current and continuous loop without capacitor for instance) using jefimenko equation as I shared in picture in terms of current density ( conductivity of wire * electric field in wire) and change of electric field multiplied by free space permittivity constant of wire, but I could not find them, could you help me about that what is the curl of time dependent magnetic field ( AC current and continuous loop without capacitor for instance)? Does it consist of current density ( conductivity of wire * electric field in wire) and change of electric field multiplied by free space permittivity constant? Could you provide me mathematical demonstration for curl of time dependent magnetic field?

 

[phyzguy]

Hi, Initially I would like express that curl of magnetic field is dependent on current density (let's think non changed current at first), but also maxwell shows that curl of B is dependent on change in electric field multiplied bu dialectic constant. My question is : If we have continuous loop/circuit (without capacitor for instance) and changed current (AC current for instance), then our curl of magnetic field is dependent on both current density ( conductivity of wire * electric field in wire) and change of electric field multiplied by free space permittivity constant of wire? Also I saw in some sites that change of electric field in wire may be negligible...

Well, let's look at it. If we look at the curl B Maxwell's equation in integral form (Ampere's law), we can write:
$$\int B \cdot dL = \mu_0 [\int J \cdot dS + \epsilon_0 \frac{\partial}{\partial t} \int E \cdot dS]$$
In a circuit, we can write $J = \sigma E$ or $E = \rho J$, where rho is the resistivity. Then we can write:
$$\int B \cdot dL = \mu_0 [\int J \cdot dS + \epsilon_0 \rho \frac{\partial}{\partial t} \int J \cdot dS] = \mu_0 [I + \epsilon_0 \rho \frac{\partial I}{\partial t} ]$$
or, for some frequency ω:
$$\int B \cdot dL = \mu_0 [I + i \epsilon_0 \rho \omega I ]$$

So the question becomes what is the magnitude of the combination ερω compared to 1. I think if you put in some numbers you will find that in any normal circuit where a reasonable current is flowing that the second term is negligible. It is only when the resistivity is so high that no significant current is flowing that the second term matters. It is hard to call this case a "circuit".

 

[mertcan]

Well, let's look at it. If we look at the curl B Maxwell's equation in integral form (Ampere's law), we can write:
$$\int B \cdot dL = \mu_0 [\int J \cdot dS + \epsilon_0 \frac{\partial}{\partial t} \int E \cdot dS]$$
In a circuit, we can write $J = \sigma E$ or $E = \rho J$, where rho is the resistivity. Then we can write:
$$\int B \cdot dL = \mu_0 [\int J \cdot dS + \epsilon_0 \rho \frac{\partial}{\partial t} \int J \cdot dS] = \mu_0 [I + \epsilon_0 \rho \frac{\partial I}{\partial t} ]$$
or, for some frequency ω:
$$\int B \cdot dL = \mu_0 [I + i \epsilon_0 \rho \omega I ]$$

So the question becomes what is the magnitude of the combination ερω compared to 1. I think if you put in some numbers you will find that in any normal circuit where a reasonable current is flowing that the second term is negligible. It is only when the resistivity is so high that no significant current is flowing that the second term matters. It is hard to call this case a "circuit".

Thank you but also may I ask you to look at my second question in post 20. Because I am aware that curl of time dependent magnetic field should depend on current density (conductivity * electric field)and change in electric field *free space permittivity but when I applied curl of jefimenko equation for time dependent magnetic field I can not obtain the desired result (current density (conductivity * electric field)and change in electric field *free space permittivity ) could you help me about that providing mathematical demonstration? ?

 

[phyzguy]

Thank you but also may I ask you to look at my second question in post 20. Because I am aware that curl of time dependent magnetic field should depend on current density (conductivity * electric field)and change in electric field *free space permittivity but when I applied curl of jefimenko equation for time dependent magnetic field I can not obtain the desired result (current density (conductivity * electric field)and change in electric field *free space permittivity ) could you help me about that providing mathematical demonstration? ?

I don't really understand your question. Maxwell's equations always hold, so we always have:

$$\nabla \times B = \mu_0 J + \frac{1}{c^2} \frac{\partial E}{\partial t}$$

whether or not B is time dependent. Given any distribution of currents and charges, you can always calculate E and B by calculating the retarded potentials, as described here. However, taking the curl of B as given there and showing that it reduces back to the right-hand side of Maxwell's equations may be a difficult exercise.