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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)?

Qmechanic
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GedankenExperimentalist
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1 Answers1

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Heald and Marion take the divergence of Faraday's and Ampere-Maxwell's laws and... [snip] ... But wouldn't this imply that there really are only two fundamental equations in classical EM (and Lorentz force law)?

No.

...And they use a similar argument for $\vec\nabla\cdot\vec E= 4\pi\rho$.

Maxwell's equations are $$ \vec \nabla \cdot \vec E = 4\pi\rho\tag{1} $$ $$ \vec \nabla \cdot \vec B = 0\tag{2} $$ $$ \vec\nabla\times\vec E = -\frac{1}{c}\frac{\partial \vec B}{\partial t}\tag{3} $$ $$ \vec\nabla\times\vec B = \frac{1}{c}\left(4\pi\vec J + \frac{\partial \vec E}{\partial t}\right)\tag{4}\;. $$

If you take the divergence of Eq. (4) you get $$ 0=4\pi\vec\nabla\cdot\vec J + \frac{\partial}{\partial t}\vec \nabla\cdot \vec E\;, $$ which is not Eq. (1).

If I am given Eq. (1) I can derive the continuity equation $$ 0=\vec \nabla\cdot\vec J + \frac{\partial\rho}{\partial t}. $$ Conversely, if I am given the continuity equation, I can derive Eq. (1). But in any case Eq. (1) does not follow from the divergence of Eq. (4).

If you take the divergence of Eq. (3) you get $$ 0=\frac{\partial}{\partial t}\vec\nabla\cdot\vec B\;, $$ which is not the same equation as Eq. (2). You may be able to make some argument that it should be effectively the same given some reasonable other assumptions, but those are still other assumptions, and are not nothing.

hft
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