Lecture 2 Electromagnetic Waves in Homogenous Media

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Lecture 2 Electromagnetic Waves in Homogenous Media. 6.013. ELECTROMAGNETICS AND APPLICATIONS. Luca Daniel. Today’s Outline. Course Overview and Motivations Maxwell Equations (review from 8.02) in integral form in differential form EM waves in homogenous lossless media EM Wave Equation

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Lecture 2Electromagnetic Wavesin Homogenous Media

6.013

ELECTROMAGNETICS AND APPLICATIONS

Luca Daniel

Today’s Outline
• Course Overview and Motivations
• Maxwell Equations (review from 8.02)
• in integral form
• in differential form
• EM waves in homogenous lossless media
• EM Wave Equation
• Solution of the EM Wave equation
• Uniform Plane Waves (UPW)
• Complex Notation (phasors)
• Wave polarizations
• EM Waves in homogeneous lossy media

Today

Maxwell’s Equations

in linear isotropic homogeneous lossless media

Constitutive

Relations

Gauss‘s

Law

0

Ampere’s Law:

0

Second derivative in space  second derivative in time,

therefore solution is any function with identical dependencies

on space and time (up to a constant)

or

What are Electromagnetic Waves

A “wave” is a fixed disturbance propagating through a medium

A,B

B

wave motion

0

z

A

A,B energy density

null

0

z

Medium A B A energy B energy

String stretch velocity potential kinetic

Acoustic pressure velocity potential kinetic

Ocean height velocity potential kinetic

Electromagnetic E H electric magnetic

Solutions of the Wave Equation

Possible solutions are many Try Uniform Plane Wave (UPW), e.g. assume:

1)

0

0

2)

3)

E+(t – z/ν)

propagation

In air/vacuum waves moves at velocity

t = t

t = 0

z

z=vt

0

0

0

0

0

Sinusoidal Uniform Plane Wave (UPW) in +z direction

General solution:

Ey(z,t) = E+(t - z/v) [V/m]

One special solution:

where

To find the magnetic field:

In air/vacuum

Note:

Uniform Plane Wave: EM fields

EM Wave in z direction:

Wavelength

x

z

y

Complex Notation (Phasors)

Complex notation for a single frequency (f = /2)

“Phasor”: contains all amplitude, vector,

spatial and phase information

UPW case

Time domain E

Example:

Phasor E

Uniform Plane Wave (UPW) in Complex Notation

x

direction of propagation

z

y

wavelength

Uniform Plane Wave (UPW) Linear vs. Circular vs. Elliptical Polarization

Linear Polarization

Circular Polarization

Image source: http://en.wikipedia.org

x

Linear Polarization

z

y