EM-waves

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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

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      Huygen's Principle

           Extra Resources

Diffraction

EM-waves

Other phenomena

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      Diffraction

           Introduction

The phenomenon of diffraction is observed when a wave is incident on an opening.

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      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

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      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

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      Polarization

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      Polarization

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      Polarization

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      Polarization

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      Polarization

\bar{S}=\bar{S}_0\ \cos^2\theta

 EM-waves

      Polarization

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      Polarization

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      Polarization

 EM-waves

      Polarization

\theta
\theta
(a)
(b)