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0

You write, "In the case of the magnetic field we are yet to observe its source or sink." If you mean "we are yet to observe a source or sink", you're correct. However, consider the magnetic vector field (ignoring units/speaking qualitatively): $$\vec{B}=(0, \frac{z}{(1+r^2)^2},\frac{y}{(1 + r^2)^2})$$ This is a valid field because it's the curl of the ...

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This become a lot clearer if you consider the integral forms of Maxwell's equations. We start with Gauss' Law $$\nabla\cdot\vec{E} = \frac{\rho}{\epsilon_0}$$ If we integrate this over some volume $V$ and apply Gauss' Divergence Theorem we find that the left hand side gives \begin{align} ...

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I) Right, the differential form of Gauss's law $$\tag{1} {\bf\nabla} \cdot{\bf E}~=~ \frac{\rho}{\varepsilon_0}$$ uses the relatively advanced mathematical concept of Dirac delta distributions in case of point charges $$\tag{2} \rho({\bf r})~=~\sum_{i=1}^n q_i\delta^3({\bf r}-{\bf r}_i).$$ Note in particular, that it is technically wrong to claim (as ...

1

Maxwell's equations state $$\nabla \cdot \vec E = \frac{\rho}{\epsilon_0}$$ $$\nabla \cdot \vec B = 0$$ If we accept Maxwell's equations as true, there is no source/sink of the magnetic field, since the divergence of the magnetic field is zero no matter what. Yet, no matter how you feel about the Dirac delta, where there is charge, there is non-zero ...

2

First of all, I'm not an expert, but that can be an advantage in trying to explain the equations in lay terms... Maxwell's equations are these, in differential form: $$\nabla \cdot \mathbf{E} = \frac {\rho} {\varepsilon_0}$$ $$\nabla \cdot \mathbf{B} = 0$$ $$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$$ $$\nabla \times ... 0 Depending on how "basic" you consider an equation to be to electromagnetism, you could consider other equations to be important enough to be thought of as basic, given the type of situation. For instance, when dealing with electromagnetism in media (typically linear media), the Constitutive Relations also apply and are necessary:$$\overrightarrow{D} = ...

3

The Maxwell equations don't need to "take into account" that the proper time of light-like paths is zero. The definition of the Minkowski metric as $$\mathrm{d}s^2 = c^2\mathrm{d}t^2 - \mathrm{d}x^2 - \mathrm{d}y^2 - \mathrm{d}z^2$$ Lorentz invariance means that all physical laws are invariant under the isometries of this metric, which are the Lorentz ...

0

The Maxwell equations only approximately describe electromagnetism, even in a pure vacuum. This is a consequence of quantum electrodynamics. One can derive corrections to the Maxwell equations; this was first done by Heisenberg an Euler in the regime where the fields only change appreciably over distances much larger than the electron Compton wavelength, see ...

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In intuitive terms, electromagnetism is the theory of the electric and magnetic fields $\vec E$ and $\vec B$. Given charge and current distribution in space, the solutions to the Maxwell equations are unique as discussed here. So, if they suffice to calculate $\vec E$ and $\vec B$ for any given electromagnetic configuration, the theory is complete. What ...

4

In electromagnetism we say that all the electromagnetic interactions are governed by the 4 golden rules of Maxwell. But I want to know that is this only an assumption It is not an assumption, it is an elegant way of joining the diverse laws of electrictity and magnetism into one mathematical framework. or a practical observation The laws of ...

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Take the wave equation $$\nabla^2\vec{E} = \frac{1}{c^2} \frac{\partial^2 \vec{E}}{\partial t^2},$$ and let $\vec{E}(\vec{r},t)$ be a solution. Indeed taking the real part $\Re(\vec{E}(\vec{r},t))$ yields the physical significant values. The initial values at $t = 0$ are $\Re(\vec{E}(\vec{r},0))$ and the problem arises here: this does not give you enough ...

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The basics about the direction of force and field comes from the "Fleming's Left Hand Rule" and the "Maxwell's Corkscrew Rule". In addition to these the Lorentz force law, i.e. F=q[E+(vxB)] gives the force on a charge moving through a magnetic and electric field [Neglect E if electric field is absent.] The Biot- Savart Law gives the relation between current ...

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You just need Maxwell's equations and the Lorentz force law. Coulombs law can be derived from Maxwell's equations.

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Going by a magic 8-ball a brief web search, the most important steps towards the geometrization of electromagnetism (ie its formulation as a classical Yang-Mills theory in terms of principal connections) should be: Maxwell's equations: James Clerk Maxwell, A dynamical theory of the electromagnetic field (1865) differential forms: Élie Cartan, Sur certaines ...

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The answer is positive. This is due to the fact that the equations describing how currents generate the field are linear. The solution is obtained by a suitable inverse of the linear operator associating currents to fields. It is fundamental to observe that this inverse operator is linear because the boundary conditions satisfy the superposition principle ...

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