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Can Maxwell's equations be used to define charge and current density? For example, we have $$ \nabla \cdot \vec{E} = 4\pi \rho $$ Can we rewrite this as $$ \rho = \frac{1}{4\pi} \nabla \cdot \vec{E} $$ and say that charge is nothing but $\frac{1}{4\pi} \nabla \cdot\vec{E}$? And could we similarly say that charge density is defined by $$ \vec{J} = \frac{1}{4\pi}\left(c \nabla\times\vec{B} - \frac{\partial \vec{E}}{\partial t}\right) $$

In other words, what's to stop us from dispensing altogether with charge and current density and thinking only in terms of electric and magnetic fields? Can we think of charge as a property of the electric field (i.e. a measure of its divergence)? Is there an experiment that suggests that charge and current density really are something distinct from the electric and magnetic fields?

Charles Hudgins
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1 Answers1

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Certainly you could do that. But it would not be very useful. You would have to already know the fields, and already knowing the fields you could then solve for or eliminate the charges and currents.

However, usually it is much easier to know the sources than the fields. So you can solve for the fields based on your knowledge of the sources. It is unclear how you would encode and use that information in a fields-first or a fields-only approach.

Dale
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