Chapter 24

Electromagnetic Waves

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X-ray Unpolarized visible light

Radio waves

Polarized visible light

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24.1 Electromagnetic Waves, Introduction Electromagnetic (EM) waves permeate

our environment EM waves can propagate through a

vacuum Much of the behavior of mechanical

wave models is similar for EM waves Maxwell’s equations form the basis of

all electromagnetic phenomena

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Conduction Current A conduction current is carried by charged

particles in a wire The magnetic field associated with this

current can be calculated by using Ampère’s Law:

The line integral is over any closed path through which the conduction current passes

od I B s

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Conduction Current, cont. Ampère’s Law in this form is

valid only if the conduction current is continuous in space

In the example, the conduction current passes through only S1 but not S2

This leads to a contradiction in Ampère’s Law which needs to be resolved

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James Clerk Maxwell 1831 – 1879 Developed the

electromagnetic theory of light

Developed the kinetic theory of gases

Explained the nature of color vision

Explained the nature of Saturn’s rings

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Displacement Current Maxwell proposed the resolution to the

previous problem by introducing an additional term called the displacement current

The displacement current is defined as

Ed o

dI

dt

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Displacement current The electric flux through S2 is

EA S2 is the gray circle A is the area of the capacitor

plates E is the electric field between

the plates If q is the charge on the

plates, then Id = dq/dt This is equal to the

conduction current through S1

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Displacement Current The changing electric field may be

considered as equivalent to a current For example, between the plates of a capacitor

This current can be considered as the continuation of the conduction current in a wire

This term is added to the current term in Ampère’s Law

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Ampère-Maxwell Law The general form of Ampère’s Law is

also called the Ampère-Maxwell Law and states:

Magnetic fields are produced by both conduction currents and changing electric fields

( ) Eo d o o o

dd I I I

dt

B s

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24.2 Maxwell’s Equations, Introduction In 1865, James Clerk Maxwell provided a

mathematical theory that showed a close relationship between all electric and magnetic phenomena

Maxwell’s equations also predicted the existence of electromagnetic waves that propagate through space

Einstein showed these equations are in agreement with the special theory of relativity

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Maxwell’s Equations Gauss’ Law (electric flux)

Gauss’ Law for magnetismFaraday’s Law of inductionAmpère-Maxwell Law

0o

B Eo o o

qd d

d dd d I

dt dt

E A B A

E s B s

The equations are for free space No dielectric or magnetic material is present

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Lorentz Force Once the electric and magnetic fields

are known at some point in space, the force of those fields on a particle of charge q can be calculated:

The force is called the Lorentz force

q q F E v B

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24.3 Electromagnetic Waves In empty space, q = 0 and I = 0 Maxwell predicted the existence of

electromagnetic waves The electromagnetic waves consist of oscillating

electric and magnetic fields The changing fields induce each other which

maintains the propagation of the wave A changing electric field induces a magnetic field A changing magnetic field induces an electric field

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Plane EM Waves We assume that the vectors

for the electric and magnetic fields in an EM wave have a specific space-time behavior that is consistent with Maxwell’s equations

Assume an EM wave that travels in the x direction with the electric field in the y direction and the magnetic field in the z direction

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Plane EM Waves, cont The x-direction is the direction of propagation Waves in which the electric and magnetic

fields are restricted to being parallel to a pair of perpendicular axes are said to be linearly polarized waves

We assume that at any point in space, the magnitudes E and B of the fields depend upon x and t only

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Equations of the Linear EM Wave From Maxwell’s equations applied to empty

space, E and B are satisfied by the following equations

These are in the form of a general wave equation, with

Substituting the values for o and o gives c = 2.99792 x 108 m/s

2 2 2 2

2 2 2 2o o o o

E E B Band

x t x t

1 o ov c

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Solutions of the EM wave equations

The simplest solution to the partial differential equations is a sinusoidal wave: E = Emax cos (kx – t)

B = Bmax cos (kx – t)

The angular wave number is k = 2 is the wavelength

The angular frequency is = 2 ƒ ƒ is the wave frequency

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Ratio of E to B The speed of the electromagnetic

wave is

Taking partial derivations also gives

2 ƒƒ

2c

k

max

max

E Ec

B k B

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Properties of EM Waves The solutions of Maxwell’s are wave-like, with

both E and B satisfying a wave equation Electromagnetic waves travel at the speed of

light

This comes from the solution of Maxwell’s equations

oo

1c

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Properties of EM Waves, 2 The components of the electric and

magnetic fields of plane electromagnetic waves are perpendicular to each other and perpendicular to the direction of propagation The electromagnetic waves are transverse

waves

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Properties of EM Waves, 3 The magnitudes of the fields in empty

space are related by the expression

This also comes from the solution of the partial differentials obtained from Maxwell’s Equations

Electromagnetic waves obey the superposition principle

BEc

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EM Wave Representation This is a pictorial

representation, at one instant, of a sinusoidal, linearly polarized plane wave moving in the x direction

E and B vary sinusoidally with x

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Rays A ray is a line along which the wave travels All the rays for the type of linearly polarized

waves that have been discussed are parallel The collection of waves is called a plane

wave A surface connecting points of equal phase

on all waves, called the wave front, is a geometric plane

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Doppler Effect for Light Light exhibits a Doppler effect

Remember, the Doppler effect is an apparent change in frequency due to the motion of an observer or the source

Since there is no medium required for light waves, only the relative speed, v, between the source and the observer can be identified

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Doppler Effect, cont. The equation also depends on the laws of

relativity

v is the relative speed between the source and the observer

c is the speed of light ƒ’ is the apparent frequency of the light seen

by the observer ƒ is the frequency emitted by the source

vc

vcff

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Doppler Effect, final For galaxies receding from the Earth, v

is entered as a negative number Therefore, ƒ’<ƒ and the apparent

wavelength, ’, is greater than the actual wavelength

The light is shifted toward the red end of the spectrum

This is what is observed in the red shift

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