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As mentioned by Ruslan , precisely speaking, what one should do to simulate the unpolarized light is to take an average of the intensity of all the orthogonal polarized light other than just 2 of them. Plane source is a special case because its z-polarized component is quite weak so it won't hurt even if only an average of x and y-polarized component is ...

6

Divergence means the field is either converging to a point/source or diverging from it. Divergence of magnetic field is zero everywhere because if it is not it would mean that a monopole is there since field can converge to or diverge from monopole. But magnetic monopole doesn't exist in space. So its divergence is zero everywhere. Mathematically, we get ...

13

Your intuition about the meaning of the divergence operator is wrong. In physics it's easiest to think intuitively about divergence by using the divergence theorem which states $$\int_V dV \ \nabla \cdot \mathbf{B} = \int_{\partial V} \mathbf{B} \cdot d\mathbf{S}$$ where $\partial V$ is the surface area surrounding the volume $V$. The magnetic field has ...

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Well, as the people said in the comments, the Theorems of Green, Stokes and Gauss will do the job, and are about as mathematically rigorous as you could hope for here! The two different sets of formula follow directly. I don't want to write all four of them out, you should be able to do them yourself, but for example, let's consider the Gauss Law. ...

2

Can we conclude that B=0? For a general field it is wrong because every constant vector will satisfy those conditions. But for the magnetic field is it enough? It depends on what facts about magnetic field you want to admit into your hypothetical situation. If you assume the Maxwell equations with vanishing sources and the condition $\nabla \times ... 2 No, that is not enough to say that$B=0$. You must also consider that $$\nabla\times E=-\frac{\partial B}{\partial t}$$ which means that for a magnetic field that is constant spatially but not in time, your conditions would be true but your$B$field would not be$0$If, however, we had a case where$\nabla\times E=0$as well, then (aside from being a very ... 0 This is easy to see if you use the Maxwell equations to arrive at the decoupled, inhomogeneous wave equations for the fields, $$\begin{split}\Box \vec{E} &= - \mu_0 \frac{\partial \vec{J}}{\partial t} - \vec{\nabla} \frac{\rho}{\varepsilon_0},\\ \Box \vec{B} &= \mu_0\vec{\nabla}\times \vec{J}, \end{split}$$ with$\Box \equiv \frac{1}{c^2} ...

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(Originally posted as a comment): The flaw is actually implicit in the statement that all electric fields are generated by a potential, $\mathbf{E}=-\nabla\phi$. In fact, that's only valid for electrostatics. Actual time-varying electric/magnetic fields are not necessarily conservative. For example, when a decreasing magnetic flux is passing through a ...

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In fact, for dynamic fields, $\vec E$ is NOT generated by a scalar potential, and instead, is generated by the equation ${\vec E} = - {\vec\nabla}\phi - \frac{\partial{\vec A}}{\partial t}$, where $\vec A$ is the magnetic vector potential, which is related to the magnetic field by ${\vec B} = {\vec \nabla} \times {\vec A}$ Of course, these relations are ...

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As Ruslan said, your error lies in the fact that you used z-polarized light. There is no such thing as z-polarized light (it doesn't exist).

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A well-known application of the duality you refer to occurs in the design of antennas, for example, magnetic vs. electric dipole antennas. A magnetic dipole is a loop antenna, an electric dipole is a linear antenna, still their radiation pattern is similar as one replaces E with H.

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The idea is that Maxwell's equations are symmetric under the exchange of certain quantities. This does not mean that you identify the electric and the magnetic field, but just say that they are dual to each other. This resolves your problem of dimensionality. If one extends this principle to the two equations which contain divergences of the fields, one ...

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