1. The equations so far..... Gauss’ Law for E Fields
Gauss’ Law for B Fields
Faraday’s Law
Ampere’s Law
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2. Ampere’s Law Current inside No current inside
Lets work with the last one…
Current inside
No current inside

3. Maxwell’s Displacement Current, Id
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4. Maxwell’s Approach
Time varying magnetic field leads to curly electric field.
Time varying electric field leads to curly magnetic field?
Current in wire I – causes change in E flux, should cause the same effect in curly B I
‘equivalent’ current
combine with current in Ampere’s law

5. Maxwell’s Equations (1865)
in Systeme International (SI or mks) units
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6. Question
Suppose you were able to charge a capacitor with constant current (does not change in time).
Does a B field exist in between the plates of the capacitor?
A) NO
B) YES
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7. Maxwell’s Equations (Free Space)
Note the symmetry of Maxwell’s Equations in free space, when no charges or currents are present
We can predict the existence of electromagnetic waves. Why? Because the wave equation is contained in these equations. Remember the wave equation.
h is the variable that is changing in space (x) and time (t). v is the velocity of the wave.
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8. Review of Waves from Mechanics
The one-dimensional wave equation:
has a general solution of the form:
A solution for waves traveling in the +x direction is:
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9. Wave Examples Electromagnetic Wave Wave on a String:
e.g., sqrt(tension/mass) is wave speed of a guitar string, proportional to frequency of fundamental
Electromagnetic Wave
What is waving?? The Electric & Magnetic Fields !!
Rewrite Maxwell’s equations as equations of the form:
The velocity of the wave, v, will be related to 0 and 0.
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10. Four Step Plane Wave Derivation
Step 1 Assume we have a plane wave propagating in z (i.e. E, B not functions of x or y)
Example:
Step 2 Apply Faraday’s Law to infinitesimal loop in x-z plane x Ex Ex x z1 z2 z By Z y
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11. Four Step Plane Wave Derivation
Step 3 Apply Ampere’s Law to an infinitesimal loop in the y-z plane: x z y z1 z2 By Z y Ex
Step 4: Use results from steps 2 and 3 to eliminate By
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12. Velocity of Electromagnetic Waves
We derived the wave equation for Ex:
The velocity of electromagnetic waves in free space is:
Putting in the measured values for 0 & 0, we get:
This value is identical to the measured speed of light! We identify light as an electromagnetic wave.
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13. Maxwell Equations: Electromagnetic Waves
Maxwell’s Equations contain the wave equation
The velocity of electromagnetic waves:
c = x 108 m/s
The relationship between E and B in an EM wave
Energy in EM waves: the Poynting vector x z y
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14. Question
If the magnetic field of a light wave oscillates parallel to a y axis and is given by By = Bm sin(kz-t) in what direction does the wave travel? -y -z y z -x
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15. Question
If the magnetic field of a light wave oscillates parallel to a y axis and is given by By = Bm sin(kz+t) in what direction does the wave travel and parallel to which axis does the associated electric field oscillate?
-z, y
z, x
-z, x
z, -x
-z, -x
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16. Electromagnetic Spectrum
~1850: infrared, visible, and ultraviolet light
were the only forms of electromagnetic waves known.
Visible light (human eye)
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17. Electro- magnetic Spectrum
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18. Wien’s Displacement Law
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19. White Light: A Mixture of Colors (DEMO)
Demos:
7C-1
7C-2
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20. Spectral Lines
Energy states of an atom are discrete and so are the
energy transitions that cause the emission of a photon
(DEMO)
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21. How is B related to E? We derived the wave equation for Ex:
We could have derived for By:
How are Ex and By related in phase and magnitude?
Consider the harmonic solution:
where
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22. E & B in Electromagnetic Waves
Plane Wave:
where: x z y
The direction of propagation is given by the cross product
where are the unit vectors in the (E,B) directions.
Nothing special about (Ex,By); eg could have (Ey, -Bx)
Note cyclical relation:
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23. Energy in Electromagnetic Waves
Electromagnetic waves contain energy. We know the energy density stored in E and B fields:
In an EM wave, B = E/c
The total energy density in an EM wave = u, where
The Intensity of a wave is defined as the average power (Pav=uav/t) transmitted per unit area = average energy density times wave velocity:
For ease in calculation define Z0 as:
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24. The Poynting Vector
The direction of the propagation of the electromagnetic wave is given by:
This energy transport is defined by the Poynting vector S as:
S has the direction of propagation of the wave
The magnitude of S is directly related to the energy being transported by the wave
The intensity for harmonic waves is then given by:
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25. Characteristics x S z
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26. Summary of Electromagnetic Radiation
combined Faraday’s Law and Ampere’s Law
time varying B-field induces E-field
time varying E-field induces B-field
E-field and B-field are perpendicular
energy density
Poynting Vector describes power flow
units: watts/m2 E B S
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