# Uniform plane wave

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Uniform plane wave. Radiation from a current filament. Spherical waves from a source of finite dimension. Transmitted waves from a finite sized source behave like spherical waves. Concept of plane waves. - PowerPoint PPT Presentation

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1Uniform plane waveEMLAB2Radiation from a current filament

EMLAB3

Spherical waves from a source of finite dimension

Transmitted waves from a finite sized source behave like spherical waves.EMLAB4

Concept of plane waves

EM waves transmitted from a finite sized source spread spherically in the space. At great distances from the source, EM waves behave as a plane wave locally.EMLAB5E & H in source free region

In source free region, E and H can be obtained easily.With J and equal to zero, a source free wave equation is obtained.

EMLAB6

If the source current away from the field point has the direction of x-axis, the electric field can have only x component. If the extent of the source is infinite along x and y directions, variations of the field along those directions become zero.

The first term represent the wave propagating toward +z direction, while the second stands for the wave propagating to the opposite direction. Considering only the wave propagating in the positive direction,

wave impedance of free spaceIf the propagating direction is in +z-axis, the Helmholtz equation becomes a simple expression.

EMLAB

7Plane wave propagating general direction

If the propagating direction is other than x, y or z direction, the phase term in the exponential function can be obtained by the inner product of the k-vector and the position vector.the k-vector can be compared to the normal vector of equi-phase plane.

EMLAB8

The magnetic field can be obtained.

Magnetic field of a plane wave

Vector identityEMLAB9Wave propagation in dielectricsIf the wave is propagating into a medium other than free space, the molecules in that medium vibrate under the action of electric field. Meanwhile, the energy of the wave is dissipated and the medium become heated.The dissipation can be modeled by a complex permittivity.

3. The ratio of the real part of r to the imaginary part is called the loss tangent of that media.

EMLAB10

EMLAB11From the equations on the left, it can be seen that the phase of displacement current leads that of conduction current by 90 degree.That is, electric field propagates first, then charges move under the action of that electric field.Helmholtz equation in a lossy medium becomes,

Propagation constant in a lossy dielectricEMLAB12

To obtain the approximate expression of and , we consider the two extreme cases of

Good dielectric : Good conductor :

EMLAB13

Case 1)

Plane wave in good dielectrics

EMLAB14

Plane wave in good conductors

is called as the skin depth of the medium at the given frequency. If the electric field penetrate into a lossy medium as much as one skin depth, its strength decreases by 1/e. That is its strength becomes 36.7% of the original value.conductor

Case 2)

EMLAB15

EMLAB16

EMLAB17ExampleFind the skin depth of sea water at the frequency of 1MHz. In sea water

good conductor

EMLAB18Example

Calculate the resistance of a round copper of 1mm radius and 1km length at DC and 1MHz.

EMLAB19Wave polarization

The Helmholtz equation has three components (x, y, z).But the divergenceless condition imposes that E = 0, which is a constraint among the vector components of E-field. That is, all the component of Ex, Ey, Ez are not independent.

From the above condition, only two components among the Ex, Ey, Ez are independent. That is, two kinds of independent polarizations comprise an arbitrary E-field.a changing direction of electric field observed at a position.Among the components of an electric field vector, only two of them is independent.EMLAB20

Polarization

Example of a electric polarization of a wave propagating in +z direction.

Linear polarizationCircular polarizationEMLAB21

Polarization diversity antennaEMLAB