I Do curl/time dependent Maxwell's equations imply divergence equations?

AI Thread Summary
The discussion centers on the implications of curl and time-dependent Maxwell's equations, particularly regarding the divergence of magnetic and electric fields. It is argued that if the magnetic field B is zero at any instant, then the divergence of B must be zero at all times, suggesting no magnetic monopoles exist. The conversation also explores whether the divergence equations are fundamental or derived from Maxwell's equations, with the conclusion that they can be seen as non-fundamental, similar to charge conservation. A counterpoint is raised regarding the existence of plane electromagnetic waves, which maintain a non-zero magnetic field at all times, challenging the assumption that B can be zero everywhere at a single instant. Ultimately, the discussion emphasizes the relationship between the divergence equations and the foundational principles of classical electromagnetism.
LightPhoton
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Just like charge conservation is not something extra, but instead is derivable from Maxwell's equations, similarly by the given argument so are the divergence equations, making them, just like conservation of charge, non-fundamental in classical EM
In Classical Electromagnetic Radiation, Heald and Marion take the divergence of Faraday's and Ampere-Maxwell's laws and state:

$$-\vec\nabla\cdot\frac{\partial\vec B}{\partial
t}=\vec\nabla\cdot\vec\nabla\times\vec E=0$$

If we assume that all the derivatives of B are continuous, we may interchange the differentiation with respect to space and time:

$$\frac{\partial}{\partial t}\vec\nabla\cdot\vec B=0$$

or

$$\vec\nabla\cdot\vec B =\text{ constant in time}$$

Now, if at any instant in time B = 0, then the constant of integration vanishes.If we assume that the field originated at some time in the past, then

$$\vec\nabla\cdot\vec B =0$$

even for the time-varying case. Physically, we may argue that there are no magnetic monopoles in the steady-state situation so that ##\vec\nabla\cdot\vec B =0##; and because the addition of time-varying fields cannot generate such monopoles, ##\vec\nabla\cdot\vec B =0## must be generally valid.


And they use a similar argument for ##\vec\nabla\cdot\vec E= 4\pi\rho##.
But wouldn't this imply that there really are only two fundamental equations in classical EM (and Lorentz force law)?
 
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Which equations are fundamental and which are derived is a matter of choice. You can also derive Maxwell’s equations from the conservation laws or from the gauge symmetry.

Also, if you do choose to start with Maxwell’s equations, then you can also derive the Lorentz force law.
 
LightPhoton said:
TL;DR Summary: Just like charge conservation is not something extra, but instead is derivable from Maxwell's equations, similarly by the given argument so are the divergence equations, making them, just like conservation of charge, non-fundamental in classical EM

In Classical Electromagnetic Radiation, Heald and Marion take the divergence of Faraday's and Ampere-Maxwell's laws and state:

$$-\vec\nabla\cdot\frac{\partial\vec B}{\partial
t}=\vec\nabla\cdot\vec\nabla\times\vec E=0$$

If we assume that all the derivatives of B are continuous, we may interchange the differentiation with respect to space and time:

$$\frac{\partial}{\partial t}\vec\nabla\cdot\vec B=0$$

or

$$\vec\nabla\cdot\vec B =\text{ constant in time}$$

Now, if at any instant in time B = 0, then the constant of integration vanishes.If we assume that the field originated at some time in the past, then

$$\vec\nabla\cdot\vec B =0$$

even for the time-varying case. Physically, we may argue that there are no magnetic monopoles in the steady-state situation so that ##\vec\nabla\cdot\vec B =0##; and because the addition of time-varying fields cannot generate such monopoles, ##\vec\nabla\cdot\vec B =0## must be generally valid.


And they use a similar argument for ##\vec\nabla\cdot\vec E= 4\pi\rho##.
But wouldn't this imply that there really are only two fundamental equations in classical EM (and Lorentz force law)?
Could you clarify how you are arguing that there must be a time at which ##\vec{B}=0## for all physical situations? Wouldn't a plane electromagnetic wave propagating in free space be a valid solution to Maxwell’s equations in Minkowski spacetime, which extends infinitely in time and has a nonzero magnetic field at every instant?

Additionally, if you must rely on physical reasoning to argue that ##\nabla \cdot \vec{B} = 0##, even if only for the steady-state case, wouldn't that imply ##\nabla \cdot \vec{B} = 0## is not being derived solely from the other Maxwell equations?
 
julian said:
Wouldn't a plane electromagnetic wave propagating in free space be a valid solution to Maxwell’s equations in Minkowski spacetime, which extends infinitely in time and has a nonzero magnetic field at every instant?
I don't follow. A linearly-polarized, monochromatic plane-wave of period ##T## propagating in free-space has ##\vec{B}=0## twice in every period.
 
renormalize said:
I don't follow. A linearly-polarized, monochromatic plane-wave of period ##T## propagating in free-space has ##\vec{B}=0## twice in every period.
The OP required ##\vec{B} = 0## everywhere in space at a single instant in time. However, a plane wave does not have ##\vec{B} = 0## everywhere at any instant—it may be zero at certain points, but it is always nonzero elsewhere. I initially had the same thought as you did.
 
The condition ##\frac{\partial}{\partial t} \vec{\nabla} \cdot \vec{B} = 0 ## implies ##\vec{\nabla} \cdot \vec{B} = f(x,y,z)##. If you require ##\vec{B} = 0## everywhere at some instant, then this forces ##f(x,y,z) = 0##, ensuring that ##\vec{\nabla} \cdot \vec{B} = 0## for all time.
 
julian said:
The OP required ##\vec{B} = 0## everywhere in space at a single instant in time. However, a plane wave does not have ##\vec{B} = 0## everywhere at any instant—it may be zero at certain points, but it is always nonzero elsewhere. I initially had the same thought as you did.
Thanks, now I understand and I agree.
 
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