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Maxwell's laws summarize the main laws of electrodynamics, which are:

- Gauss's law for electric fields

- Gauss's law for magnetic fields

- Faraday's law of electrical induction

- Ampère's law of magnetic induction

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ID:(821, 0)

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In the case where we have a vacuum there are neither charges nor currents, so Maxwell's equations are reduced to

\vec{

abla}\cdot\vec{E}=0

\vec{

abla}\cdot\vec{B}=0

\vec{

abla}\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}

\vec{

abla}\times\vec{B}=\frac{1}{c^2}\frac{\partial\vec{E}}{\partial t}

If we derive the fourth in time and use the third, we obtain

\frac{1}{c^2}\frac{\partial^2\vec{E}}{\partial t^2}=\vec{

abla}\times\frac{\partial\vec{B}}{\partial t}=-\vec{

abla}\times\vec{

abla}\times\vec{E}

and if the third is derived in time and uses the fourth

\frac{\partial^2\vec{B}}{\partial t^2}=-\vec{

abla}\times\frac{\partial\vec{E}}{\partial t}=-c^2\vec{

abla}\times\vec{

abla}\times\vec{B}

How

\vec{

abla}\times\vec{

abla}\times\vec{a}=\vec{

abla}(\vec{

abla}\cdot\vec{a})-

abla^2\vec{a}

and considering the first two Maxwell equations, the wave equations are obtained

\frac{1}{c^2}\frac{\partial^2\vec{E}}{\partial t^2}=

abla^2\vec{E}

\frac{1}{c^2}\frac{\partial^2\vec{B}}{\partial t^2}=

abla^2\vec{B}

If the wave propagates along the z axis the solution of the equations is of the form

\vec{E}(z,t)=\vec{E}_0e^{i(\omega t-kz)}

\vec{B}(z,t)=\vec{B}_0e^{i(\omega t-kz)}

where $ k $ is the wave vector, $ \ omega $ the angular frequency and there is no dependence on the axes perpendicular to the direction of propagation. On the other hand, the requirement that the divergence of the fields is zero leads to the absence of components of either the electric or magnetic fields in the direction of wave propagation.

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ID:(282, 0)