Question about Ampere's law in vacuum and in matter

In summary, the conversation discusses the derivations of Maxwell's equations in vacuum and in matter. It is mentioned that for materials where the assumption that D is proportional and parallel to E is not true, the regular Maxwell equations may not work. The solution is found by considering the total current density as the sum of free current density, magnetization current density, and polarization current density.
  • #1
Arham
26
0
Hi

We can derive equation [itex]\nabla.D=\rho_f[/itex] from equation [itex]\nabla.E=\rho/\epsilon_0[/itex]. But what about Ampere's law? I tried to derive [itex]\nabla\times{H}=J_f+\partial{D}/\partial{t}[/itex] from [itex]\nabla\times{B}=\mu_0J+\epsilon_0\mu_0\partial{E}/\partial{t}[/itex] but I could not. This is strange because I thought that Maxwell's equations in vacuum are enough for studying electromagnetic field in any matter and that Maxwell's equations in matter are derivable from them.
 
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  • #2
They are, if you add some assumptions about the material - D proportional (and parallel) to E and so on.
For materials where this is not true, I don't know.
 
  • #3
[itex]\partial{D}/\partial{t}=\epsilon_0\partial{E}/\partial{t}+\partial{P}/\partial{t}[/itex]. The second term is underivable from Ampere's law in vacuum.
 
  • #4
Add the assumption that ##D \propto E##, and it works.

In general, this can be wrong, but I don't know if the regular Maxwell equations work there at all. If ##\epsilon_r## is a tensor (or nonlinear), things can get difficult.
 
  • #5
Dear mfb,

I think I found the solution. [itex]\partial{P}/\partial{t}[/itex] is some kind of current (bound charges are moving). So if we write total current density as [itex]J=J_f+\nabla\times{M}+J_p[/itex] where [itex]J_p[/itex] is polarization current density, we can solve the problem.
 
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1. What is Ampere's law?

Ampere's law is a fundamental law in electromagnetism that describes the relationship between electric currents and magnetic fields. It states that the line integral of the magnetic field around a closed loop is equal to the permeability of free space times the current passing through the loop.

2. How does Ampere's law apply to vacuum?

In a vacuum, Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the product of the permeability of free space and the current passing through the loop. This relationship holds true as long as there are no changing electric fields present.

3. How does Ampere's law apply to matter?

In matter, Ampere's law is modified to include the effect of magnetic materials. The line integral of the magnetic field around a closed loop is equal to the product of the permeability of the material and the current passing through the loop, as well as the magnetization of the material. This accounts for the additional magnetic field created by the alignment of magnetic dipoles in the material.

4. What is the significance of Ampere's law in practical applications?

Ampere's law is used in many practical applications, such as in the design of electrical circuits, motors, and generators. It helps engineers and scientists understand and predict the behavior of magnetic fields in these systems, allowing for efficient and effective designs.

5. Are there any limitations to Ampere's law?

Ampere's law is based on certain assumptions, such as the absence of changing electric fields and the linear relationship between the magnetic field and current. In situations where these assumptions do not hold true, such as in high-frequency circuits or in the presence of strong electric fields, Ampere's law may not accurately predict the behavior of magnetic fields.

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