em waves 1

8/3/2019 em waves 1

1/26

Physics 1051

Lecture 24

Electromagnetic Waves

8/3/2019 em waves 1

2/26

Recommended Problems

Fishbane, Chapter 34, Problems 7,10,14,16,21,24,30

8/3/2019 em waves 1

3/26

The story so far

So far we have talked about oscillations and waves,

electric forces and fields, and magnetic forces and

fields.

In the next couple of lectures we use all of this

material to develop a description ofelectromagnetic

waves.

8/3/2019 em waves 1

4/26

What do we know about

electricity and magnetism? Charged particles produce electric fields

Moving charges produce magnetic fields

Electric fields cause a force on a charged particle

Magnetic fields cause a force on moving charges

particles

8/3/2019 em waves 1

5/26


electricity and magnetism? The electric flux through a closed surface is

proportional to the charge inside the surface

(Gausss Law).

This says that electric fields are due to electric

charges. Electric field lines begin and end atcharges. We can use it to show that the electric

field of a point charge is proportional to 1/r2.

0

Q E dA

I !

&&

8/3/2019 em waves 1

6/26


electricity and magnetism? The magnetic flux through a closed surface is zero.

This says that there are no magnetic monopoles,

and that magnetic field lines have no beginning or

end.

0B

dA !

&&

8/3/2019 em waves 1

7/26


electricity and magnetism? Integrating the magnetic field around a closed path gives a

result that is proportional to the current enclosed by the

path. (Ampres Law)

This says that magnetic fields are caused by currents.

There is another part to this that we have not discussed:Magnetic fields can also be caused by changing electric

fields. Including this mathematically would give us another

term on the rhs that included the time derivative of a surface

integral of the electric field. I will spare you the details.

0 encB ds IQ !& &

8/3/2019 em waves 1

8/26


electricity and magnetism? Changed in magnetic flux cause an induced emf.

(Faradays Law).

This says that changing magnetic fields produce an

electric potential, or in other words changing

magnetic fields produce an electric field.

surface

d

dt

d E ds B dA

dt

I *!

! && &&

8/3/2019 em waves 1

9/26


electricity and magnetism? These laws show that there is a strong connection

between electricity and magnetism. In factelectricity and magnetism are two aspects of thesame phenomenon, which we can callelectromagnetism.

The electric and magnetic fields are coupled theydepend on each other.

In a vacuum, if we have no electric charges and noelectric currents, then the equations weve writtendown above are perfectlysymmetric with respect to

Eand B. This means thatEand Bbehave the same.

8/3/2019 em waves 1

10/26

Maxwells Equations

The equations weve written down here are a form of

Maxwells Equations.

Maxwell saw that all of these electromagnetic

phenomena could be described in a single

mathematical framework.

The solution to Maxwells equations in a vacuum

turns out to be traveling waves

both the electric andmagnetic field propagate through space in the form of

waves.

8/3/2019 em waves 1

11/26


8/3/2019 em waves 1

12/26

0

0

( , ) sin

( , ) sin

E r t E kx t

B r t B kx t

[

[

!

!

& &&

& &&

k!2T

P

[ !2T

T

! 2Tf

The plane wave solutions to Maxwells Equations have the

following form

where E0 and B0 are theamplitudes of the electric and

magnetic components of the wave.

kis the wave number of the electromagnetic waves. (Really it

should be a vector pointing in the direction of propagation ofthe wave.)

k! wave number

P = wave length

f ! frequency

8/3/2019 em waves 1

13/26

Maxwells equations tell us several things:

The electric and magnetic fields propagate as traveling

waves

These waves travel at the speed of light. The speed of the

waves is given by

The electric and magnetic fields are always perpendicular to

each other.

The amplitudes of the fields are also related:

0 0

1v c

Q I! !

0 0 0E B !

& &

E cB!

8/3/2019 em waves 1

14/26


Maxwells equations, published in 1864, had far reaching

consequences. They showed in detail how electricity and

magnetism were connected. Maxwells equations predicted the

existence of electromagnetic waves, which traveled with the

speed of light. Maxwell conjectured that light was a form ofelectromagnetic radiation.

In 1887 Heinrich Hertz

produced radio waves and

from the measured frequency

and wavelength confirmed

Maxwells conjecture.

8/3/2019 em waves 1

15/26

Electromagnetic Spectrum

Radio Waves

Microwaves

Infrared

Visible light

Ultraviolet

X-rays Gamma rays

8/3/2019 em waves 1

16/26

Example 1

Fishbane, Chapter34, Problem 5

If the electric field for a plane electromagnetic wave

is given by

what are and the direction of propagation of the

wave?

0

0

cos( )

x

y

E

E E kz t [

!

!

B&

8/3/2019 em waves 1

17/26

Example 2

Fishbane, Chapter34, Problem 61.

Use dimensional analysis to show that

has the dimensions of speed.

0 0

1

Q I

8/3/2019 em waves 1

18/26

Electromagnetic Energy

Electromagnetic waves carry energy and can transport

energy from one place to another. The energy carried

by an electromagnetic waves is shared equallybetween the magnetic field part of the wave and the

electric field part of the wave. The average value of

the total energy density in the wave is

220 0

0

0

1

2 2

Eu B

I

Q! !

8/3/2019 em waves 1

19/26

Intensity and Energy Transport

c

c

x

R

Since electromagnetic fields

represent a form of energy,

and electromagnetic waves

propagate through space, then

electromagnetic waves

transport energy.

Consider a segment of acircular beam of circular light

of radius R. How much

energy is transported across a

surface in time (t?

8/3/2019 em waves 1

20/26

8/3/2019 em waves 1

21/26

Electromagnetic Energy

The power per unit area delivered by an

electromagentic wave to a surface perpendicular to the

direction of propagation is

This is called the intensity of the em wave.

220 0 0 0

0

0 02 2 2

c E EBc I cu B

I

Q Q! ! ! !

8/3/2019 em waves 1

22/26

Example

The intensity of sunlight on the

earths surface is approximately

1 kW/m2.

Calculate the energy density of

sunlight at the earths surface.

Estimate the amplitude of the

electric and magnetic fields of

the sunlight striking the earth.

u !I

c

!10

3

3v

10

8 J m-3

! 3.3v106 J m-3

I! uc

8/3/2019 em waves 1

23/26

u }I0E0

2

2

To calculate the amplitude of

the electric field we use the

relation

E}2u

I0

}2 v 3.3 v106

8.85v1012V m

-1

} 864 V m -1

Thus we obtain

E0

B0

! c

To calculate the amplitude of

the magnetic field we use the

relation

B0 !E0

c

! 8643v108

T

! 2.87 v106 T

Thus we obtain

8/3/2019 em waves 1

24/26

Example 3

A typical laser pointer has a power of around 0.5 mW

and a beam width of around 1mm.

Calculate the intensity of the beam.

Calculate the amplitude of the electric and magnetic

fields of the beam.

8/3/2019 em waves 1

25/26

Production of EM Waves

Electromagnetic waves

are produced by creating

an oscillating electric or

magnetic field.

Charges oscillating in a

linear antenna will produce

an oscillating electric field.

An oscillating electric field

will produce an oscillating

magnetic field.

This produces a wave which

travels away from the source

with the speed of light.

8/3/2019 em waves 1

26/26

There are other types of

antenna. For example the loop

antenna, uses an oscillating

current to produce a oscillating

magnetic field.

Documents

em waves 1