# B Does the first Maxwell equation work also for moving charges?

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1. Jun 8, 2017

### Cozma Alex

My professor said that the first two maxwell equation's are about static fields

Another fast question:

Do a single charge that is moVing generate a magnetic field?
I saw a demonstration that explains the magnetic field as an electric field using only special relativity, there is a free charge outside a wire, it is moving relative to the wire, there is current in a wire, a current is made with multiple charges and the lorentz contraction due to the relative motion makes the density on charges different, so there is like more electrons than protons according to the free moving charge in a section of the wire...
But how can this demonstration be done if there is no wire but just a single charge moving?
So there is our free charge and another charge moving...
There is magnetic field?
How can you contract a Charge? It's without dimension

2. Jun 8, 2017

### Paul Colby

Unless I missed the memo there is no standard numbering of Maxwell's equations. Also, the complete equations hold for all cases static and time dependent. So,

$\nabla\cdot D = \rho$​

holds always including the dynamic case.

I gather what you mean by this is charges are modeled as points how can they contract? Well, the fields they produce do in the sense that a field of a stationary charge is a radial directed E-field. This viewed by another in relative motion to the charge is both an E field and a B field.

3. Jun 10, 2017

### vanhees71

The full microscopic Maxwell equations read
$$\vec{\nabla} \cdot \vec{E}=\rho, \quad \vec{\nabla} \times \vec{B}-\frac{1}{c} \partial_t \vec{E}=\frac{1}{c} \vec{j}$$
and
$$\vec{\nabla} \cdot \vec{B}=0, \quad \vec{\nabla} \times \vec{E}+\frac{1}{c} \partial_t \vec{B}=0.$$
These are valid for all cases, no matter whether it's a static, stationary, quasistationary or fully general time-dependent problem. How you number the Maxwell Equations is arbitrary, by the way. I've chosen the above order, because it follows the physical meaning: The first two describe the generation of the electromagnetic field out of the sources, which are charge and current densities. The latter two are constraints ensuring that you have socalled gauge theory, i.e., they admit the introduction of the four-potential up to a four-gradient. One consequence is that from the Maxwell equations alone you can derive charge conservation, i.e.,
$$\partial_t \rho + \vec{\nabla} \cdot \vec{j}=0,$$
without the need to use the dynamical equations of matter, which have to be added to Maxwell's equations to make it a complete closed dynamical system.