Download Chapter 22 Lecture Notes 1.1 Changing Electric Fields Produce

Chapter 22 Lecture Notes
Physics 2424 - Strauss
Formulas:
ΣB ∆l = µ0I + µ0ε0∆Φ E/∆t
Φ E = EA
λ = c/f
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c=
ε 0 µ0
S = P/A = U/tA = uV /tA = uc
u = U/V = (1/2)ε0E2 + 1/(2µ0)B2 =ε0E2 = (1/µ0)B2
1. ELECTROMAGNETIC WAVES
1.1 Changing Electric Fields Produce Magnetic Fields
We have learned in chapter 21 that changing magnetic fields create electric
fields. In the mid-1800’s, the famous Scottish physicist James Clerk
Maxwell hypothesized that changing electric fields might also create
magnetic fields and that this interaction would create electromagnetic (EM)
waves, waves of oscillating electric and magnetic fields. He predicted that
the speed of these electromagnetic waves would be the speed of light.
Consequently, he predicted that visible light was an electromagnetic wave.
Let’s look at the first part of his theory, that changing electric fields
produce magnetic fields.
Your book describes how Maxwell hypothesized something called the
displacement current, which was shown to be equivalent to a changing
electric flux. That is
ID = ε0∆Φ E/∆t
where the electric flux (Φ E) is defined in the same way as the magnetic
flux, the amount of electric field going through an area.
Φ E = EA
Maxwell showed that with this definition, Ampere’s law had an additional
term and should be written.
ΣB ∆l = µ0I + µ0ε0∆Φ E/∆t
Recall that Ampere’s law stated in mathematical terms that a current
produced a magnetic field. With this additional term, Maxwell showed that
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a changing electric flux would create a magnetic field, as well. So now we
have the ideas that a changing electric flux creates a magnetic field, and a
changing magnetic flux creates an electric field. Maxwell realized that this
would lead to the production and self-propagation of electromagnetic
waves. Let’s see how this works.
1.2 Production of EM Waves
Let’s assume we have an antenna, which is just some kind of wire that is
connected to an ac source. The ac source produces oscillating + and charges which set up electric field (due to the separation of charge) and a
magnetic field (due to the current in the wire).
+
ac source
×
×
E field is the arrow
B field is the crosses
Note that the electric and magnetic fields are perpendicular to each other.
This field begins to move away from the antenna and in a little while the ac
source has caused the situation to reverse.
•
•
+
We have a magnetic field that is oscillating in and out of the paper and an
electric field that is oscillating up and down in the paper. Since a changing
magnetic field creates an electric field and a changing electric field creates
a magnetic field, these two oscillating fields continue to reinforce each
other, and the wave propagates through space.
We have shown how to create an EM waves using an antenna. In general,
electromagnetic waves are created by any accelerating charge.
1.3 The Nature of EM Waves
What are the properties of these electromagnetic waves?
1.3.1 TRANSVERSE WAVE
They are transverse waves. That means the direction of oscillation is
perpendicular to the direction of motion. The electric and magnetic fields
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are sinusoidally oscillating perpendicular to the direction of motion. See
figure 22-7.
1.3.2 COMPOSED OF ELECTRIC AND MAGNETIC FIELDS
The waves are composed of electric and magnetic fields. As we have seen,
1.
2.
3.
4.
The fields are oscillating
The fields are at right angles to each other
The fields are at right angles to direction of motion
The fields are in phase. The peek of the magnetic field occurs at the
same time as the peek of the electric field.
Again, this can be seen from figure 22-7
1.4 The Speed of EM Waves
One amazing thing about Maxwell’s hypothesis was that he predicted what
the speed of electromagnetic radiation should be. The book shows briefly
how to calculate this from principles that we already know. Using a
similar method, Maxwell showed that the speed of electromagnetic waves
in a vacuum would be
c=(ε0µ0)-1/2
c= 1/[(8.85×10-12C2/(N-m2)(4π×10-7T-m/A)]1/2 = 3.00×108 m/s2
We check that the units are okay.
F = ILB ⇒ N = C⋅m⋅T/s T = N⋅s/C⋅m
Amps = C/s, so the units above are
{(N⋅m2/C2)(C⋅m/N⋅s)(1/m)(C/s)}1/2 = {m2/s2}1/2 = m/s
This is amazing. He predicted that the speed of light would be exactly what
we measure it to be. Galileo had tried to measure the speed of light using
lantern’s and an assistant at a distant location. A more accurate way of
measuring the speed of light is using a rotating mirror.
light source
d
mirror
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The eight sided mirror rotates, and the eye can only see the light from the
box when the mirror rotates exactly 1/8 of a revolution in the time it takes
the light to bounce off the mirror and come back. Since the time it takes to
make one rotation is given by the period, T = 1/f = 2 π /ω , one eighth of a
rotation takes a time of t =π/4ω,
Or v = d/t = 4d ω /π So by knowing the angular rotation ( ω ) and the
distance (d) precisely, one can measure the speed of light precisely.
The speed of light is FAST. It is 3.00×108 m/s2 or 186,000 miles/second.
Problem: The sun is about 1.5 × 1011 m from the earth. How long does it
take light to get here?
1.5 The Spectrum of EM Waves
Electromagnetic waves come in a variety of wavelengths that appear very
different to our senses or detectors, but are all electromagnetic waves.
These include radio waves (AM, FM), television signals, microwaves,
infrared waves (heat), visible light, ultraviolet rays (cause skin cancer), xrays, gamma rays. Because these are all electromagnetic waves,
1. They have the same speed.
2. They have different frequency and wavelength
Recall from our discussion of waves in 2414 that the velocity of a wave is
given by
v = c = λ f.
So if I know the frequency, I can determine the wavelength, and vice versa.
I show one characteristic frequency, when really everything is a range.
Name
Extra Low Freq
Audio Frequency
Radio Frequency
Microwave
Infrared (Heat)
Visible
Frequency
60 Hz
10 kHz (1×104)
222 MHz (2×108)
10 GHz (1×1010)
10 THz (1×1013)
600 Thz (6×1014)
Wavelength (λ)
5000 km (5×106)
30 km (3×104)
1.4 m
30 mm (3×10-2)
30 µm (3×10-5)
500 nm (5×10-7)
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Time for one λ
17 ms (1.7×10-2)
100 µs (1×10-4)
4.5 ns (4.5×10-9)
100 ps (1×10-10)
100 fs (1×10-13)
1.7 fs (1.7×10-15)
Ultraviolet
X-ray
Gamma-ray
1×1016 Hz
1×1018 Hz
1×1020 Hz
30 nm (3×10-8)
.1 fs (1×10-16)
300 pm (3×10-10) 1×10-18 s
3 pm (3×10-12)
1×10-20 s
1.6 The Energy of EM Waves
We have seen in our study of electricity and magnetism that there is energy
stored in a magnetic field and in an electric field. They have energy
densities.
ue = (1/2)ε0E2
um = (1/2µ0)B2
Because em waves are oscillating electric and magnetic fields, they, too
carry energy. In an em wave, the total energy density is just the sum of the
contributions from the electric and magnetic waves.
u = (1/2)ε0E2 + (1/2µ0)B2
The energy carried in the magnetic field is the same as the energy in the
electric field.
(1/2)ε0E2 = (1/2µ0)B2 so E = cB
So we can write density as u = ε0E2 or u = (1/µ0)B2
As always with sinusoidal waves
ERMS = (1/√2)E0 and BRMS = (1/√2)B0
We often measure the energy stored in an em wave by measuring its
intensity, which is the total power that passes through a unit area. (This
heat is intense. Your body is the area). (S is the intensity)
S = P/A = U/tA
measured in watts/m2
S = uV /tA (V is volume = Al = Avt = Act)
S = uAct/tA = uc
S = (1/2)cε0E2 + (c/2µ0)B2 or S = cε0E2 or S = (c/µ0)B2
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Problem: A light bulb has an RMS power output of 100 watts. (a) If I am
standing 2 meters from the bulb, how much energy do I feel on my face in
one minute if my face has an area of 0.02 m2? (b) What is the rms magnetic
field incident on my face?
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