Электромагнитные колебания

Материал из Википедии — свободной энциклопедии

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Электромагнитные колебания можно изобразить в виде самораспространяющихся поперечных колебаний электрического и магнитного полей. На рисунке - плоскополяризованная волна, распространяющаяся справа налево. Колебания электрического поля изображены в вертикальной плоскости, а колебания магнитного поля — в горизонтальной.

Электромагнитными колебаниями называются периодические изменения напряженности Е и индукции В.

Электромагнитными колебаниями являются свет, радиоволны, микроволны, инфракрасное излучение, ультрафиолетовое излучение, рентгеновские лучи, гамма-лучи.

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Эта статья содержит фрагменты на английском языке.

Вы можете помочь проекту, переведя её до конца

БЛА БЛА БЛА БЛАБЛА БЛАБ ЛА БАЛАЛАЛУОАЩШГУРПГШЩУРШГПЦРГУПРШГЦelectricity and magnetism, known as Maxwell's equations. If you inspect Maxwell’s equations without sources (charges or currents) then you will find that, along with the possibility of nothing happening, the theory will also admit nontrivial solutions of changing electric and magnetic fields. Beginning with Maxwell’s equations for free space:

$\nabla \cdot \mathbf{E} = 0 \qquad \qquad \qquad \ \ (1)$

$\nabla \times \mathbf{E} = -\frac{\partial}{\partial t} \mathbf{B} \qquad \qquad (2)$

$\nabla \cdot \mathbf{B} = 0 \qquad \qquad \qquad \ \ (3)$

$\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial}{\partial t} \mathbf{E} \qquad \ \ \ (4)$

где

$\nabla$ — векторный дифференциальный оператор (набла).

Одно решение,

$\mathbf{E}=\mathbf{B}=\mathbf{0}$ ,

— простейшее.

Чтобы найти другое, более интересное решение, we utilize vector identities, which work for any vector, as follows:

$\nabla \times \left( \nabla \times \mathbf{A} \right) = \nabla \left( \nabla \cdot \mathbf{A} \right) - \nabla^2 \mathbf{A}$

To see how we can use this take the curl of equation (2):

$\nabla \times \left(\nabla \times \mathbf{E} \right) = \nabla \times \left(-\frac{\partial \mathbf{B}}{\partial t} \right) \qquad \qquad \qquad \quad \ \ \ (5) \,$

Evaluating the left hand side:

$\nabla \times \left(\nabla \times \mathbf{E} \right) = \nabla\left(\nabla \cdot \mathbf{E} \right) - \nabla^2 \mathbf{E} = - \nabla^2 \mathbf{E} \qquad \quad \ (6) \,$

where we simplified the above by using equation (1).

Evaluate the right hand side:

$\nabla \times \left(-\frac{\partial \mathbf{B}}{\partial t} \right) = -\frac{\partial}{\partial t} \left( \nabla \times \mathbf{B} \right) = -\mu_0 \epsilon_0 \frac{\partial^2}{\partial^2 t} \mathbf{E} \qquad (7)$

Equations (6) and (7) are equal, so this results in a vector-valued differential equation for the electric field, namely

{|cellpadding="2" style="border:2px solid #ccccff"

| $\nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} \mathbf{E}$ |}

Applying a similar pattern results in similar differential equation for the magnetic field:

{|cellpadding="2" style="border:2px solid #ccccff"

| $\nabla^2 \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} \mathbf{B}$ . |}

These differential equations are equivalent to the wave equation:

$\nabla^2 f = \frac{1}{{c_0}^2} \frac{\partial^2 f}{\partial t^2} \,$

where

c₀ is the speed of the wave in free space and

f describes a displacement

Or more simply:

$\Box^2 f = 0$

where $\Box^2$ is d’Alembertian:

$\Box^2 = \nabla^2 - \frac{1}{{c_0}^2} \frac{\partial^2}{\partial t^2} = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} - \frac{1}{{c_0}^2} \frac{\partial^2}{\partial t^2} \$

Notice that in the case of the electric and magnetic fields, the speed is:

$c_0 = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$

Which, as it turns out, is the speed of light in free space. Maxwell’s equations have unified the permittivity of free space $ε 0$ , the permeability of free space $μ 0$ , and the speed of light itself, c₀. Before this derivation it was not known that there was such a strong relationship between light and electricity and magnetism.

But these are only two equations and we started with four, so there is still more information pertaining to these waves hidden within Maxwell’s equations. Let’s consider a generic vector wave for the electric field.

$\mathbf{E} = \mathbf{E}_0 f\left( \hat{\mathbf{k}} \cdot \mathbf{x} - c_0 t \right)$

Here $\mathbf{E}_0$ is the constant amplitude, $f$ is any second differentiable function, $\hat{\mathbf{k}}$ is a unit vector in the direction of propagation, and ${\mathbf{x}}$ is a position vector. We observe that $f\left( \hat{\mathbf{k}} \cdot \mathbf{x} - c_0 t \right)$ is a generic solution to the wave equation. In other words

$\nabla^2 f\left( \hat{\mathbf{k}} \cdot \mathbf{x} - c_0 t \right) = \frac{1}{{c_0}^2} \frac{\partial^2}{\partial^2 t} f\left( \hat{\mathbf{k}} \cdot \mathbf{x} - c_0 t \right)$ ,

for a generic wave traveling in the $\hat{\mathbf{k}}$ direction.

This form will satisfy the wave equation, but will it satisfy all of Maxwell’s equations, and with what corresponding magnetic field?

$\nabla \cdot \mathbf{E} = \hat{\mathbf{k}} \cdot \mathbf{E}_0 f'\left( \hat{\mathbf{k}} \cdot \mathbf{x} - c_0 t \right) = 0$

$\mathbf{E} \cdot \hat{\mathbf{k}} = 0$

The first of Maxwell’s equations implies that electric field is orthogonal to the direction the wave propagates.

$\nabla \times \mathbf{E} = \hat{\mathbf{k}} \times \mathbf{E}_0 f'\left( \hat{\mathbf{k}} \cdot \mathbf{x} - c_0 t \right) = -\frac{\partial}{\partial t} \mathbf{B}$

$\mathbf{B} = \frac{1}{c_0} \hat{\mathbf{k}} \times \mathbf{E}$

The second of Maxwell’s equations yields the magnetic field. The remaining equations will be satisfied by this choice of $\mathbf{E},\mathbf{B}$ .

Not only are the electric and magnetic field waves traveling at the speed of light, but they have a special restricted orientation and proportional magnitudes, $E 0 = c 0 B 0$ , which can be seen immediately from the Poynting vector. The electric field, magnetic field, and direction of wave propagation are all orthogonal, and the wave propagates in the same direction as $\mathbf{E} \times \mathbf{B}$ .

From the viewpoint of an electromagnetic wave traveling forward, the electric field might be oscillating up and down, while the magnetic field oscillates right and left; but this picture can be rotated with the electric field oscillating right and left and the magnetic field oscillating down and up. This is a different solution that is traveling in the same direction. This arbitrariness in the orientation with respect to propagation direction is known as polarization.

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Электромагнитные колебания

Материал из Википедии — свободной энциклопедии

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