EM-waves
more on EM-waves
Energy & Intensity
EM-waves
Energy
The energy stored in the electric field in the volume between the plates of a parallel plate capacitor is given by
EM-waves
Energy & Intensity
How much energy does an EM wave carry?
U=\tfrac{1}{2}CV^2
=\tfrac{1}{2}\frac{\epsilon_0 A}{d}E^2d^2
V=\int E\ ds=E\ d
C=\frac{\epsilon_0 A}{d}
=\tfrac{1}{2}\epsilon_0E^2Ad
The energy density stored in the electric field:
EM-waves
Energy & Intensity
How much energy does an EM wave carry?
u=U/\text{volume}
u=\frac{1}{2}\epsilon_0E^2
The energy density stored in the magnetic field:
EM-waves
Energy & Intensity
How much energy does an EM wave carry?
u=\frac{1}{2\mu_0}B^2
EM-waves
Energy & Intensity
How much energy does an EM wave carry?
For an electromagnetic wave, for a sample volume, which field carries more share of the energy (E or B) ?
EM-waves
Energy & Intensity
How much energy does an EM wave carry?
u=\frac{1}{2\mu_0}B^2
u=\frac{1}{2}\epsilon_0E^2
The energy carried by the wave per unit area per unit time is called the energy flux S:
EM-waves
Energy & Intensity
How much energy does an EM wave carry?
S=\frac{1}{\mu_0}EB
Taking the direction of propagation into account, we get the Poynting vector:
\vec{S}=\frac{1}{\mu_0}\vec{E}\times\vec{B}
For an electromagnetic wave given by:
EM-waves
Energy & Intensity
How much energy does an EM wave carry?
E(t)=E_0 \sin(\omega t)
We define the intensity as the average flux density over one cycle
I=S_\text{avg}=\frac{c}{2}\epsilon_0E_0^2
=\frac{c}{2 \mu_0}B_0^2
B(t)=B_0 \sin(\omega t)
EM-waves
Energy & Intensity
How much energy does an EM wave carry?
A plane electromagnetic wave travels northward. At one instant, its electric field has a magnitude of 6.0 V/m and points eastward. What are the magnitude and direction of the magnetic field at this instant?
B(x_0,t_0)=E(x_0,t_0)/c
=\frac{6.0\ V/m}{3\times10^8\ m/s}
=2\times10^-8\ T
The electric field, the magnetic field, and the Poynting vectors are mutually orthogonal. Curl from E (east) to B (downwards), to get S (north)
EM-waves
Energy & Intensity
How much energy does an EM wave carry?
The beam from a small laboratory laser has a radius of 2.0mm and a power of 15.0 mW. Assuming that the beam is composed of plane waves, calculate the amplitudes of the electric and magnetic fields in the beam.
EM-waves
Energy & Intensity
How much energy does an EM wave carry?
A light bulb emits 5.00 W of power as visible light. What are the electric and magnetic fields from the light at a distance of 3.0 m?
EM-waves
Energy & Intensity
How much energy does an EM wave carry?
A 150-W lightbulb emits 5% of its energy as electromagnetic radiation. What is the magnitude of the average Poynting vector 10 m from the bulb?
Wave Interference
EM-waves
Other phenomena
Suppose that two sources are generating two waves with similar properties:
EM-waves
Interference
Introduction
The waves arriving from multiple sources will have different phases:
y_1(x,t)=Y_1 \sin(\omega t-kx)
y_2(x,t)=Y_2 \sin(\omega t-kx)
y_1(x,t)=Y_1 \sin(\omega t-\phi_1)
y_2(x,t)=Y_2 \sin(\omega t-\phi_2)
Case I: the two waves arrive in phase
EM-waves
Interference
Introduction
Case II: the two waves arrive out of phase
Case I: the two waves arrive in phase
EM-waves
Interference
Introduction
Case II: the two waves arrive out of phase
Case I: the two waves arrive in phase
EM-waves
Interference
Introduction
Case II: the two waves arrive out of phase
Huygen's principle
EM-waves
Other phenomena
EM-waves
Huygen's Principle
Introduction
The Dutch scientist Christiaan Huygens (1629–1695) developed a useful technique for determining in detail how and where waves propagate.
Huygens’s principle states that every point on a wave front is a source of wavelets that spread out in the forward direction at the same speed as the wave itself. The new wave front is tangent to all of the wavelets.
EM-waves
Huygen's Principle
Textbook examples
EM-waves
Huygen's Principle
Textbook examples
Forward propagation of plane waves
EM-waves
Huygen's Principle
Textbook examples
Outward propagation of spherical waves
EM-waves
Huygen's Principle
Textbook examples
Reflection of a plane wave by a plane mirror
EM-waves
Huygen's Principle
Textbook examples
Diffraction through an opening
EM-waves
Huygen's Principle
Extra Resources
Diffraction
EM-waves
Other phenomena
EM-waves
Diffraction
Introduction
The phenomenon of diffraction is observed when a wave is incident on an opening.
EM-waves
Diffraction
Single slit diffraction
When the wavelength of the incident wave is comparable to the size of the opening, a single slit diffraction pattern is observed
EM-waves
Diffraction
Single slit diffraction
Analysis
(a) All rays are in phase; big bright peak at 0
(b) For each ray in the top half of the opening, there is a ray from the bottom half of the opening that destructively interferes with it.
\Delta l = \lambda/2
EM-waves
Diffraction
Single slit diffraction
destructive interference for a single slit
a\ \sin\theta = m \lambda
for
m=\pm 1, \pm 2, \pm 3, \dots
EM-waves
Diffraction
Single slit diffraction
EM-waves
Diffraction
Double slit diffraction
The double-slit pattern is a product of the diffraction and the interference patterns
EM-waves
Diffraction
Babinet's Principle
Babinet's principle states that the diffraction pattern from an opaque body is identical to that from a hole of the same size and shape except for the overall forward beam intensity.
i.e. the diffraction pattern from a thin strip is the same as that from a thin slit.
EM-waves
Diffraction
[CA] Single slit diffraction
Find the thickness of a human hair.
Step 1.
The Doppler Effect
EM-waves
Other phenomena
f_o=f_s\left(1\pm\frac{v_\text{relative}}{c}\right)
v_\text{relative}<<<{c}
The Doppler Effect for Sound Waves
Polarization
EM-waves
Other phenomena
EM-waves
Polarization
EM-waves
Polarization
EM-waves
Polarization
EM-waves
Polarization
EM-waves
Polarization
EM-waves
Polarization
\bar{S}=\bar{S}_0\ \cos^2\theta
EM-waves
Polarization
EM-waves
Polarization
EM-waves
Polarization
EM-waves
Polarization
\theta
\theta
(a)
(b)
Electromagnetic Waves - II
By drmoussaphysics
