11/6/171
Maxwell Equations: Electromagnetic Waves
Maxwell’s Equations contain the wave equation
The velocity of electromagnetic waves:
c = 2.99792458 x 108 m/s
The relationship between E and B in an EM wave
Energy in EM waves: the Poynting vector
x
z
y
11/6/172
The equations so far.....
Gauss’ Law for E Fields Gauss’ Law for B Fields
Faraday’s Law Ampere’s Law
11/6/173
A problem with Ampere’s Law•Time dependent situation: current flows in the wire as the capacitor charges up or down.
Consider a wire and a capacitor. C is a loop.
11/6/174
Maxwell’s Displacement Current, Id
11/6/175
Calculating Displacement CurrentConsider a parallel platecapacitor with circular plates of radius R. If charge is flowing onto one plate and off the other plate at a rate I = dQ/dt what is Id ?
•The displacement current is not a current. It represents magnetic fields generated by time varying electric fields.
11/6/176
Calculating the B fieldExample
11/6/177
Maxwell’s Equations (1865)
in Systeme International (SI or mks) units
11/6/178
Maxwell’s Equations (Free Space)Note the symmetry of Maxwell’s Equations in free space, when no charges or currents are present
h is the variable that is changing in space (x) and time (t). v is the velocity of the wave.
We can predict the existence of electromagnetic waves. Why? Because the wave equation is contained in these equations. Remember the wave equation.
11/6/17 9
Review of Waves from MechanicsThe one-dimensional wave equation:
has a general solution of the form:
A solution for waves traveling in the +x direction is:
11/6/1710
Four Step Plane Wave Derivation
Example:
Step 2 Apply Faraday’s Law to infinitesimal loop in x-zplane
Step 1 Assume we have a plane wave propagating in z(i.e. E, B not functions of x or y)
x
z
y
z1 z2
Ex Ex
∆Z
∆x
By
11/6/1711
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
∆yBy
Ex
Step 4: Use results from steps 2 and 3 to eliminate By
11/6/1712
Velocity of Electromagnetic WavesWe derived the wave equation for Ex:
The velocity of electromagnetic waves in free space is:
Putting in the measured values for µ0 & ε0, we get:
This value is identical to the measured speed of light! We identify light as an electromagnetic wave.
11/6/1713
Electromagnetic Spectrum
~1850: infrared, visible, and ultraviolet lightwere the only forms of electromagnetic waves known.
Visible light (human eye)
11/6/1714
Electro-magneticSpectrum
Sunscreen Absorbs UV
11/6/1715
http://upload.wikimedia.org/wikipedia/commons/thumb/0/0d/UV_and_Vis_Sunscreen.jpg/800px-UV_and_Vis_Sunscreen.jpg
11/6/1716
What a Bee Sees:
11/6/1717
Wien’s Displacement Law
11/6/1718
How is B related to E?
where
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:
11/6/1719
E & B in Electromagnetic WavesPlane Wave:
where:
x
z
y
where are the unit vectors in the (E,B) directions.
The direction of propagation is given by the cross product
Nothing special about (Ex,By); eg could have (Ey, -Bx)Note cyclical relation:
11/6/1720
Energy in Electromagnetic WavesElectromagnetic waves contain energy. We know the
energy density stored in E and B fields:
The Intensity of a wave is defined as the average power (Pav=uav/Δt) transmitted per unit area = average energy density times wave velocity:
• For ease in calculation define Z0 as:
u Ec
E uB E= = =12
12
2
20
02
µεIn an EM wave, B = E/c
The total energy density in an EM wave = u, where
11/6/1721
The Poynting VectorThe direction of the propagation of the
electromagnetic wave is given by:This energy transport is defined by the Poynting
vector S as:
The intensity for harmonic waves is then given by:
S has the direction of propagation of the waveThe magnitude of S is directly related to the energy being transported by the wave
11/6/1722
Characteristics
Sxz
11/6/1723
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
BS
