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