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

[LightPhoton]

TL;DR

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)?

 

[Dale]

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.

 

[julian]

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?

 

[renormalize]

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.

 

[julian]

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.

 

[julian]

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.

 

[renormalize]

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.