Download - Electromagnetic Waves Physics 4 Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB.

Electromagnetic Waves

Physics 4

Prepared by Vince Zaccone

For Campus Learning Assistance Services at UCSB

Maxwell’s Equations



Maxwell’s equations summarize the relationships between electric and magnetic fields. A major consequence of these equations is that an accelerating charge will produce electromagnetic radiation.

∮𝐸 ∙𝑑 �⃗�=𝑄𝑒𝑛𝑐𝑙

𝜖0𝑮𝒂𝒖𝒔 𝒔′ 𝒔𝑳𝒂𝒘 𝒇𝒐𝒓 �⃗�

∮ �⃗� ∙𝑑 �⃗�=0𝑮𝒂𝒖𝒔 𝒔′ 𝒔 𝑳𝒂𝒘 𝒇𝒐𝒓 �⃗�

∮𝐸 ∙𝑑 �⃗�=−𝑑Φ𝐵

𝑑𝑡𝑭𝒂𝒓𝒂𝒅𝒂𝒚 ′ 𝒔𝑳𝒂𝒘

∮ �⃗� ∙𝑑�⃗�=𝜇0(𝑖𝑐+𝜖0 𝑑Φ𝐸

𝑑𝑡 )𝑒𝑛𝑐𝑙

𝑨𝒎𝒑𝒆𝒓 𝒆 ′ 𝒔𝑳𝒂𝒘

Electromagnetic (EM) waves can be produced by atomic transitions (more on this later), or by an alternating current in a wire. As the charges in the wire oscillate back and forth, the electric field around them oscillates as well, in turn producing an oscillating magnetic field.

We have a right-hand-rule for plane EM waves:

1) Point the fingers of your right hand in the direction of the E-field

2) Curl them toward the B-field.

3) Stick out your thumb - it points in the direction of propagation.

Electromagnetic Waves



Click here for an EM wave animation

http://micro.magnet.fsu.edu/primer/java/polarizedlight/emwave/index.html

Like any other wave, we know the relationship between the wavelength and frequency, and the speed of propagation of the wave:

fvwave




fvwave

In the case of EM waves, it turns out that the wave speed is the speed of light.

So our formula for EM waves (in vacuum) is:

smcfc 8

00

1031

;




fvwave

In the case of EM waves, it turns out that the wave speed is the speed of light.

So our formula for EM waves (in vacuum) is:



smcfc 8

00

1031

;

The speed of light is also related to the strengths of the Electric and Magnetic fields.

E=cB (in standard metric units)

The continuum of various wavelengths and frequencies for EM waves is called the Electromagnetic Spectrum






• Find the frequency of blue light with a wavelength of 460 nm.




• Find the frequency of blue light with a wavelength of 460 nm.

Hz105.6m10460

103cffc 14

9sm8




• A cell phone transmits at a frequency of 1.25x108 Hz. What is the wavelength of this EM wave?




• A cell phone transmits at a frequency of 1.25x108 Hz. What is the wavelength of this EM wave?

m4.2Hz1025.1

103

fc

fc8sm8

Field of a Sinusoidal Wave

Electromagnetic waves must satisfy the WAVE EQUATION:

In the case of EM waves, both the electric and magnetic fields need to satisfy this equation. Solving this equation yields formulas for the E and B fields.

In particular, here are formulas for the E and B fields associated with a sinusoidal EM plane wave propagating in the +x-direction:



𝜕2 𝑦 (𝑥 ,𝑡)𝜕 𝑥2

= 1𝑣2

𝜕2 𝑦 (𝑥 ,𝑡)𝜕𝑡2

�⃗� (𝑥 , 𝑡 )=𝐸𝑚𝑎𝑥 cos (𝑘𝑥−𝜔𝑡) �̂�

�⃗� (𝑥 , 𝑡 )=𝐵𝑚𝑎𝑥cos (𝑘𝑥−𝜔𝑡) �̂�

Notice that these fields are perpendicular to each other, as well as the propagation direction. A right hand rule comes in handy to remember the directions.

𝑘=𝑤𝑎𝑣𝑒𝑛𝑢𝑚𝑏𝑒𝑟=2𝜋𝜆

𝜔=𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦=2𝜋 𝑓







𝜕2 𝑦 (𝑥 ,𝑡)𝜕 𝑥2

= 1𝑣2

𝜕2 𝑦 (𝑥 ,𝑡)𝜕𝑡2

�⃗� (𝑥 , 𝑡 )=𝐸𝑚𝑎𝑥 cos (𝑘𝑥−𝜔𝑡) �̂�





Example: A sinusoidal EM wave of frequency 6.10x1014Hz travels in vacuum in the +z-direction. The B-field is parallel to the y-axis and has amplitude 5.80x10-4T.

Write the equations for the E and B fields.







𝜕2 𝑦 (𝑥 ,𝑡)𝜕 𝑥2

= 1𝑣2

𝜕2 𝑦 (𝑥 ,𝑡)𝜕𝑡2

�⃗� (𝑥 , 𝑡 )=𝐸𝑚𝑎𝑥 cos (𝑘𝑥−𝜔𝑡) �̂�





Example: A sinusoidal EM wave of frequency 6.10x1014Hz travels in vacuum in the +z-direction. The B-field is parallel to the y-axis and has amplitude 5.80x10-4T.

Write the equations for the E and B fields.

�⃗� (𝑧 , 𝑡 )=𝐸𝑚𝑎𝑥 cos (𝑘𝑧−𝜔𝑡)�̂�

�⃗� (𝑧 , 𝑡 )=𝐵𝑚𝑎𝑥cos (𝑘𝑧−𝜔𝑡) �̂�

𝐸𝑚𝑎𝑥=𝑐𝐵𝑚𝑎𝑥=(3 ∙108 𝑚𝑠 ) (5.8 ∙10− 4𝑇 )=1.74 ∙105 𝑉𝑚

𝜔=2𝜋 (6.1 ∙1014𝐻𝑧 )=3.83 ∙1015 𝑟𝑎𝑑𝑠

𝑘=𝜔𝑐

=3.83 ∙1015

𝑟𝑎𝑑𝑠

3∙108𝑚𝑠

=1.28 ∙107𝑟𝑎𝑑𝑚

EM Waves in matter

So far we have assumed that electromagnetic waves propagated through empty space. If they travel through a transparent material medium (glass, air, water, etc.) the speed of propagation changes.



This is the speed in vacuum𝑐=1

√𝜖0𝜇0

𝑣=1

√𝜖𝜇=

1

√𝐾𝜖0𝐾𝑚𝜇0=

𝑐

√𝐾 𝐾𝑚

=𝑐𝑛

This is the speed in a material medium with dielectric constant* K and relative permeability Km

For most materials Km is close to one, so we can effectively ignore it and get

𝑛=√𝐾 𝐾𝑚≈√𝐾 n is called the index of refraction for the medium

Since K>1, the speed of an EM wave in a material medium is always less than c.

*K is not technically a constant – when rapidly oscillating fields are present the value is usually smaller than with constant fields, so the value of K is dependent on the frequency of the EM wave.

Energy and momentum in EM Waves

Electromagnetic waves transport energy. The energy associated with a wave is stored in the oscillating electric and magnetic fields.

We will find out later that the frequency of the wave determines the amount of energy that it carries. Since the EM wave is in 3-D, we need to measure the energy density (energy per unit volume).



This is the energy per unit volume𝑢=12𝜖0𝐸

2+12𝜇0

𝐵2=𝜖0𝐸2


Electromagnetic waves transport energy. The energy associated with a wave is stored in the oscillating electric and magnetic fields.

We will find out later that the frequency of the wave determines the amount of energy that it carries. Since the EM wave is in 3-D, we need to measure the energy density (energy per unit volume).



This is the energy per unit volume𝑢=12𝜖0𝐸

2+12𝜇0

𝐵2=𝜖0𝐸2

The Poynting vector describes the energy flow rate.

𝑆=1𝜇0

�⃗�× �⃗�

This vector usually oscillates rapidly, so it makes sense to talk about the average value, which turns out to be the INTENSITY of the radiation, with units W/m2.

For a sinusoidal wave in vacuum we can write this in several forms:

𝐼=𝑆𝑎𝑣=𝐸𝑚𝑎𝑥 𝐵𝑚𝑎𝑥

2𝜇0=𝐸𝑚𝑎𝑥

2

2𝜇0𝑐=12𝜖0𝑐𝐸𝑚𝑎𝑥

2



Example: High-Energy Cancer TreatmentScientists are working on a technique to kill cancer cells by zapping them with ultrahigh-energy pulses of light that last for an extremely short amount of time. These short pulses scramble the interior of a cell without causing it to explode, as long pulses do.

We can model a typical such cell as a disk 5.0 µm in diameter, with the pulse lasting for 4.0 ns with a power of 2.0x1012 W. We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse.

a) How much energy is given to the cell during this pulse?

b) What is the intensity (in W/m2) delivered to the cell?

c) What are the maximum values of the electric and magnetic fields in the pulse?

Recall that power is energy/time. So 2.0x1012 W is 2.0x1012 Joules/sec.



J8000J108)s100.4()100.2(Energy 39sJ12

This is the total energy, which is spread out over 100 cells, so the energy for each individual cell is 80 Joules.






To get intensity, we need to divide power/area. The area for a cell is just the area of a circle:



211262 m100.2)m105.2(rArea






To get intensity, we need to divide power/area. The area for a cell is just the area of a circle:



211262 m100.2)m105.2(rArea

Now divide to get intensity:

2mW21

29

12

2100.1

m100.2

W100.2

r100

PowerIntensity

This is the total area of all 100 cells.






To get the field strengths, recall our intensity formula:



𝐼=12𝜖0𝑐𝐸𝑚𝑎𝑥

2






𝐸𝑚𝑎𝑥=√ 2 𝐼𝜖0𝑐=√ 2∙1021

(8.85 ∙10−12)(3 ∙108)=8.68 ∙1011

𝑉𝑚

𝐵𝑚𝑎𝑥=𝐸𝑚𝑎𝑥

𝑐=2.89 ∙103𝑇

EM waves also carry momentum. This means that a ray of light can actually exert a force. To get the pressure exerted by a sinusoidal EM wave, just divide the intensity by the speed of light.



This is the same as the total energy absorbed by the surface.

If the energy is reflected, the pressure is doubled.


𝑅𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒=𝑆𝑎𝑣

𝑐



Example: Solar SailsSuppose a spacecraft with a mass of 25,000 kg has a solar sail made of perfectly reflective aluminized film with an area of 2.59x106 m. If the spacecraft is launched into earth orbit and then deploys its sail at right angles to the sunlight, what is the acceleration due to sunlight? Assume that at the earth’s distance from the sun, the intensity of sunlight is 1410 W/m2.



Recall that Pressure = Force/Area. We can use this and F=ma to get our formula:

mF

aamF

APFAF

P

mAP

a

2

2

sm4

4

26

mN6

1072.9kg105.2

m1059.2)107.4(2a

Since the sunlight reflects from our solar sail we should double the given pressure.

Example: Solar SailsSuppose a spacecraft with a mass of 25,000 kg has a solar sail made of perfectly reflective aluminized film with an area of 2.59x106 m. If the spacecraft is launched into earth orbit and then deploys its sail at right angles to the sunlight, what is the acceleration due to sunlight? Assume that at the earth’s distance from the sun, the intensity of sunlight is 1410 W/m2.

𝑅𝑎𝑑𝑖𝑎𝑡𝑖𝑜𝑛 𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒=𝑆𝑎𝑣

𝑐=1410

𝑊𝑚2

3 ∙108𝑚𝑠

=4.7 ∙10− 6𝑃𝑎

When EM waves are reflected we can have a superposition of waves traveling in opposite directions, forming a STANDING WAVE. After combining the formulas for the opposite-directed waves, and applying a bit of trigonometry, we arrive at formulas for the E and B fields of a standing EM wave.



Standing EM Waves

𝐸𝑦 (𝑥 , 𝑡 )=−2𝐸𝑚𝑎𝑥sin (𝑘𝑥 )𝑠𝑖𝑛(𝜔𝑡)

𝐵𝑧 (𝑥 , 𝑡 )=−2𝐵𝑚𝑎𝑥 cos (𝑘𝑥 )𝑐𝑜𝑠(𝜔𝑡)

We can find the positions where these fields go to zero (at all times t). These are called the NODAL PLANES:

For the E-field we need sin(kx)=0, which leads to the following locations:

𝑥=0 ,𝜆2,𝜆 ,

3𝜆2,2𝜆 ,…

Similarly for the B-field we need cos(kx)=0, which gives:

𝑥=𝜆4,3𝜆4,5𝜆4,…

If we have 2 reflecting surfaces parallel to each other we can “trap” a standing EM wave in a box, just like having a standing wave on a stretched string. The formulas are even the same:



Standing EM Waves

𝜆𝑛=2𝐿𝑛

(𝑛=1,2,3 , ..)

These formulas give the wavelengths and frequencies for standing waves that will “fit” in a box of length L

𝑓 𝑛=𝑛𝑐2𝐿

(𝑛=1,2,3 ,..)