Law conservation of charge relativistic

In summary, the law of conservation of charge relativistic states that the total charge in a closed system remains constant over time, regardless of any physical transformations or interactions that may occur. This principle is fundamental to relativity, as it shows that the total charge of a system is invariant in all inertial frames of reference. It has been experimentally verified and is considered a universal law that applies to all physical processes. The law of conservation of charge relativistic is closely related to other conservation laws, such as the conservation of energy and momentum. It is a key principle that helps to explain and predict the behavior of charged particles in various systems.
  • #1
Petar Mali
290
0
[tex]\frac{\partial \rho}{\partial t}+div\vec{j}=0[/tex]

In Deckart coordinate system

[tex]\frac{\partial j_x}{\partial x}+\frac{\partial j_y}{\partial _y}+\frac{\partial j_z}{\partial _z}+\frac{\partial (c\rho)}{\partial (ct)}=0[/tex]

definition

[tex]divA^{\mu}=\frac{\partial A^{\mu}}{\partial x^{\mu}}[/tex]
scalar (invariant)

Why I define divergence like that? Is there some certain rules for that?

[tex]j^{\mu}=(j_x,j_y,j_z,c\rho)=(\vec{j},c\rho)[/tex]

[tex]\frac{\partial j^{\mu}}{\partial x^{\mu}}=0[/tex]

[tex]divj^{\mu}=0[/tex]

Now is satisfied

[tex]j_{\mu}=(-\vec{j},c\rho)[/tex]

Can I interprate this like time inversion. Changing od indeces, think of that?

What is with

[tex]divj_{\mu}=?[/tex]
 
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  • #2
I'm not sure what you're asking, but if charge is conserved then [itex]\frac{\partial \rho}{\partial t}=0[/itex] and so [itex]\nabla j_\mu=0[/itex]. Physically this means that the amount of charge entering a volume of space must be equal to the amount leaving the volume. Or, expressed as a surface integral, the current crossing the boundary of the volume is zero. Hence

[tex]
\nabla j_\mu=0 \rightarrow \nabla( -j_\mu)=0
[/tex]

just means that reversing the current leaves the divergence unchanged. If [itex]\frac{\partial \rho}{\partial t}\ne 0[/itex] then reversing the current changes the sign of [itex]\frac{\partial \rho}{\partial t}[/itex].

See Stoke's theorem.
 
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  • #3
You cannot obtain the conservation of charge simply by proposing that

[tex]
j^{\mu}=(j_x,j_y,j_z,c\rho)=(\vec{j},c\rho)
[/tex]

is a perfectly good contravariant vector in four dimensions. And it is.

You're on the right track. It just doesn't lead to conservation of charge. You need to combine Ampere's law and Gauss's law to get charge conservation. Normally it is called the "charge continuity equation" that you might find in wikipedia.

(Mentz114, I'm still thinking over your posts in another thread, thinking of something intelligent to add.)
 
  • #4
My question is bold in the text and also after them is very interesting sign ?
 
  • #5
@ Phrak

[tex]
divj^{\mu}=0
[/tex]

This is law conservation of charge. I don't see why it wasn't finished?
 
  • #6
You should drop the dot product convention and adhere to valid tensor operations in the four dimensions of spacetime.

With this, in simplest form, the charge continuity equation is

[itex]D_\mu J^\mu = 0[/itex]

where
[tex]D_\mu = (\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}, -\frac{\partial}{c\partial t})[/itex]
and
[tex]J^\mu = (j_x, j_y, j_z, c\rho)[/itex]

The result is a scalar. It could be any scalar, but it is zero because both charge and current are derivatives of the electric and magnetic fields. It happens to be zero because partial derivatives commute.

[tex] \frac{\partial}{\partial x^i} \frac{\partial}{\partial x^j} -
\frac{\partial}{\partial x^j} \frac{\partial}{\partial x^i}=0[/itex]

You can find all of this is wikipedia somewhere where you would find that J is a derivative of the Faraday tensor.
 
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  • #7
I don't understand. Ask again.
 
  • #8
I have problems with definition of divergence and curl (rot)

[tex]
divA^{\mu}=\frac{\partial A^{\mu}}{\partial x^{\mu}}
[/tex]

[tex]rotA_{\mu}=\frac{\partial A_{\nu}}{\partial x^{\mu}}-\frac{\partial A_{\mu}}{\partial x^{\nu}}[/tex]

Why I define this like that?

In one mathematical physics book I find

[tex](rotA_i)_j=\frac{\partial A_i}{\partial x^j}-\frac{\partial A_j}{\partial x^i}[/tex]
 
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  • #9
Yes, divergence and curl are very confusing operators at first. In more advanced mathematics they finally make sense.

They make more sense understood as a dot product and a cross product.

The partials,

[itex]\nabla = \left( \frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z} \right)[/itex]
is a vector in three dimensions.

A vector such as

[itex]\left( A\frac{\partial}{\partial x},B\frac{\partial}{\partial y},C\frac{\partial}{\partial z} \right)[/itex]
obeys all the rules of a vector space over the field of real numbers.
 
  • #10
Petar Mali said:
@ Phrak

[tex]
divj^{\mu}=0
[/tex]

This is law conservation of charge. I don't see why it wasn't finished?

Now I understand. Yes, this is conservation of charge.
 
  • #11
Mentz114 said:
I'm not sure what you're asking
Petar Mali said:
My question is bold in the text and also after them is very interesting sign ?
Petar Mali, when someone points out that your question is confusing it will be more productive to clarify your intention rather than to point out some trivial formatting. For example the following question:
Petar Mali said:
Changing od indeces, think of that?
Is indeed in bold and does have a question mark afterwards, but it is gramattically incorrect and meaningless despite the presence of the correct formatting. It is not that your questions could not be found, but that they could not be understood.
 

Related to Law conservation of charge relativistic

1. What is the law of conservation of charge relativistic?

The law of conservation of charge relativistic states that the total charge in a closed system remains constant over time, regardless of any physical transformations or interactions that may occur.

2. How does the law of conservation of charge relativistic apply to relativity?

The law of conservation of charge relativistic is a fundamental principle of relativity, as it shows that the total charge of a system is invariant in all inertial frames of reference. This means that the amount of charge in a closed system will remain the same, regardless of the observer's frame of reference.

3. Is the law of conservation of charge relativistic always true?

Yes, the law of conservation of charge relativistic is a fundamental law of physics and has been experimentally verified in a wide range of systems and scenarios. It is considered to be a universal law that applies to all physical processes.

4. Can the law of conservation of charge relativistic be violated?

No, the law of conservation of charge relativistic is a fundamental law of physics and has never been observed to be violated. It is a key principle that helps to explain and predict the behavior of charged particles in various systems.

5. How does the law of conservation of charge relativistic relate to other conservation laws?

The law of conservation of charge relativistic is closely related to other conservation laws, such as the conservation of energy and momentum. These laws are all fundamental principles that govern the behavior of physical systems and are interconnected in various ways.

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