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Chapter 3

Electromagnetic & Maxwell’s Equations Part I Maxwell’s Equations Maxwell

(13 June 1831 – 5 November 1879) was a Scottish physicist.

Famous equations published in 1861 Maxwell’s Equations: Integral Form

Gauss's law

Gauss's law for

Faraday's law of induction (Maxwell–Faraday equation)

Ampère's law (with Maxwell's addition) Maxwell’s Equations

Relation of the speed of light and electric and magnetic vacuum constants 1 c  = 2.99792458  108 [m/s] 0  0

of free space, also called the electric constant

permeability of free space, also called the magnetic constant Gauss’s Law

For any closed surface enclosing total charge Qin, the net electric through the surface is

This result for the is known as Gauss’s Law. Magnetic Gauss’s Law The net through any closed surface is equal to zero:

As of today there is no evidence of magnetic monopoles See: Phys.Rev.Lett.85:5292,2000 Ampère's Law The in space around an is proportional to the electric current which serves as its source:    B  ds  0I

I is the total current inside the loop.   ds  B

i1 Direction of integration i 3 i2 Faraday’s Law

The change of magnetic flux in a loop will induce emf, i.e.,    B   E  ds    dA A t

Lenz's Law

Claim: Direction of induced current must be so as to oppose the change; otherwise would be violated. Problem with Ampère's Law

Maxwell realized that Ampere’s law is not valid when the current is discontinuous.

When charge a parallel plate capacitor:

For path L:    B  ds  0I For path R:    B  ds  0

He concluded that when the charge within an enclosed surface is changing, it is necessary to add to Ampere’s law another current called the Displacement current: ID

Then the Ampere’s law is changed to:   B  ds   ( I  I )  0 D   E wikimedia.org  0I  0 0   dA A t 11 Maxwell’s Equations: Integral Form   Q  E  dA  Gauss's law    0 B  dA  0 Gauss's law for magnetism      B Faraday's law of induction  E  ds  A  dA (Maxwell–Faraday equation) t     E Ampère's law (with  B  ds  0I  00   dA A t Maxwell's addition) Differential Operators  A Ay A the divergence operator A x   z  Div (“del”) x  y  z

xˆ y ˆ zˆ the operator     A   curl, rot x  y  z Ax A y A z

 the partial derivative with respect to time  t Other notation used  x   x Transition from Integral to Differential Form Gauss’ theorem for a vector field     F  dA  V  FdV

Volume V, surrounded by surface A

Stokes' theorem for a vector field

    A  F  ds  A  F  dA L

Surface A, surrounded by contour L Maxwell’s Equations: Differential Form

  Divergence of electric field is a function   E  of (Gauss Law) 0   B A closed loop of E field lines will exist when  E   the magnetic field varies with time t (Faraday’s Law)    B  0 Divergence of magnetic field =0 (closed loops) (Gauss Law)    E A closed loop of B field lines will exist in  B  0 j  00 The presence of a current and/or t time varying electric field (Ampere’s Law)

where  is the charge density, and s Part II Electromagnetic Maxwell’s Equation in Free Space

Since a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field, once sinusoidal fields are created they can propagate on their own. These propagating fields are called electromagnetic waves. Oscillating Electric Consider static electric field produced by an electric dipole (a) Positive (negative) charge at the top (bottom) (b) Negative (positive) charge at the top (bottom) Now imagine these two charge are moving up and down and exchange their position at every half-period. Then between the two cases there is a situation like as shown in figure below:

What is the electric field in the blank area?

Static electric fields Oscillating Electric Dipole

Since we don’t assume that change propagate instantly once new position is reached the blank represents what has to happen to the fields in meantime. We learned that E field lines can’t cross and they need to be continuous except at charges. Therefore a plausible guess is as shown in the right figure. Oscillating Electric Dipole What actually happens to the fields based on a precise calculate is shown in Fig. Magnetic fields are also formed. When there is electric current, magnetic field is produced. If the current is in a straight wire circular magnetic field is generated. Its magnitude is inversely proportional to the distance from the current. Maxwell’s Equation in Free Space Since a changing electric field produces a magnetic field, and a changing magnetic field produces an electric field, once sinusoidal fields are created they can propagate on their own. These propagating fields are called electromagnetic waves. Maxwell’s Equation in Free Space

In free space  = 0 and = 0, thus    B   E  0  E   t    E   B  0  B   00 t In Cartesian coordiantes, E Ey E  x   z  0   E  0 x y z  B By B   B  0 x   z  0 x y z Maxwell’s Equation in Free Space

E Ey B z    x  y z t  B E E By  E   x  z   t z x t Ey E B  x   z x y t

Bz By Ex   00  y z t  E Bx Bz Ey  B  00   00 t z x t By Bx Ez   00 x y t Maxwell’s Equation in Free Space

Apply the curl identity   (  )   (  )   2 to Ampere’s law    B  B  E    (  E )   t t     (   E ) 2E   (  B ) t    E Since   E  0 and   B   0  0 , one has t   2 2 2 2 2  E 2     E  00     t 2 x2 y2 z2 Maxwell’s Equation in Free Space Similarly, apply the curl identity   (  )   (  )   2 to Faraday’s law, one has   2 2  B  B  00 t 2 These two equations are plane wave equations 2 2 2 2  Ex  Ex  Ex  Ex    00 x2 y2 z2 t 2   2 2 2 2 2 2  E  Ey  Ey  Ey  Ey  E  00    00 t 2 x2 y2 z2 t 2 2 2 2 2  Ez  Ez  Ez  Ez    00 x2 y2 z2 t 2 Maxwell’s Equation in Free Space The general equation is a wave equation

2 2 2 1 2    x2 y2 z2 v2 t 2

And v is the speed of the wave. Thus, for the EM wave obtained from Maxwell’s equations, one has 1 v  00

Speed of light! Part III Plane Wave Plane Transverse Wave

If there is only one component of E-field:

One component has:

Ey B   z x t E The induced E-field and B- field perpendicular to each other. The wave equation B becomes 2 2  Ey  Ey  00 x2 t 2 Plane Transverse Wave

The solution for the wave equation is a plane wave 2 2  Ey  Ey  00 x2 t 2

and

c where   2f Angular Frequency   Wavelength f 2  T 1/ f  2 / Period k  Wavenumber  c Speed  k Energy Density Carried by the Plane Wave

Electric energy density Magnetic energy density 2 B 2  0Ey u  z uE  B 2 20 2 Ey c 00 2 1 2 uB   Ey  0Ey 20 20 2 For an EM wave, the instantaneous electric and magnetic energies are equal. 2 1 2 Bz uE  uB  0Ey  2 20

Total energy density of an EM Wave 2 2 Bz u  uE  uB   0 Ey  0 2 2   0 E0 cos (t  kx ) Energy Density Carried by the Plane Wave

Since the light frequency is high (~ 1014 Hz), when measure the energy density, the detector usually obtain its time average:

2 u   0 Ey T T For a harmonic function f(t), / / T is the period of the harmonic function

 t / 2 2 u  t /  0E0 cos (t  kx )dt T 2 2 1 2 B0   0E0  2 20

Energy Carried by the Plane Wave Poynting vector : the flow of electromagnetic energy along the wave propagation direction, i.e., the transport of energy per unit time across a unit area

EM wave speed c, in t time across area A, l  ct

uctA S   uc tA A thus 1 S  Ey Bz 0 Considering the energy flows in x-direction, S should be a vector  1   S  E  B 0 Energy Carried by the Plane Wave The of the EM wave, or the light is the time average of the magnitude of the Poynting vector

c 0 2 I  S  E0 T 2 or

I  S T  cuav

The intensity is c times the total average energy density Spherical Wave Another type of transversal wave is called spherical wave

A spherical wave is a constant- frequency wave whose wavefronts (surfaces of constant phase) are parallel concentric spheres of constant amplitude normal to the phase velocity vector.

When the distance from the source is very large, a spherical wave can be locally approximated as a plane wave. Spherical Wave The wave amplitude is , as it propagates further from the source, the amplitude decreases inversely with respect to r.

The (W/m2) (intensity) of the wave is proportional to , That is a familiar inverse square law of propagation for spherical wave disturbance. At r = 0, wave is not valid but rather describes a point source.

Over a small enough region (or sufficiently far away from the source), the spherical wavefronts associated with a spherical wave are approximately planar. That is, waves emanating from point sources can be adequately described by plane waveforms when the region of interest is small compared to the distance from the point source. Part IV

Particle Nature of Light Models of Light

Light is a wave Light is a particle

So which one is right?

•They are both right...and they are both wrong. •That’s called wave-particle duality •In some experiments, the wave model works best. •In other experiments, the particle model works best. •Thus, we use both. Blackbody Radiation Photoelectric Effect Hertz J.J. Thomson When UV light is shone on a metal plate in a vacuum, it emits charged particles (Hertz 1887), which were later shown to be electrons by J.J. Thomson (1899).

Results: Maximum KE of ejected electrons is independent of intensity, but dependent on ν

For ν<ν0 (i.e. for frequencies below a cut-off frequency) no electrons are emitted There is no time lag. However, rate of ejection of electrons depends on light intensity. Photoelectric Effect

Classical expectations Electric field E of light exerts force F=-eE on electrons. As intensity of light increases, force increases, so KE of ejected electrons should increase.

Electrons should be emitted whatever the frequency ν of the light, so long as E is sufficiently large

For very low intensities, expect a time lag between light exposure and emission, while electrons absorb enough energy to escape from material Photoelectric Effect Einstein Einstein’s interpretation (1905): Light comes in packets of energy (photons)

E h Millikan

An electron absorbs a Verified in detail through subsequent single photon to leave experiments by Millikan the material

The maximum KE of an emitted electron is then

Kmax  hW  Work function: minimum Planck constant: energy needed for electron to universal constant of escape from metal (depends on nature material, but usually 2-5eV) h 6.63  1034 Js Photoelectric Effect Photon Properties Relation between particle and wave properties of light Photon Energy and frequency E h

If the light intensity is I, the average number of photons impinging on a unit area per unite time J (mean photon flux density) is

The mean photons flux Photon Properties

Relativistic formula relating 2 2 2 2 4 energy and E pc  mc

For light E pc and c  

h h p    c Photon Properties EM wave exert a P on a matter 2 1 2 Bz uE  uB  0Ey  2 20 Average pressure: S I P  T  T c c For a single photon, the momentum is: nhv P  T c Thus the momentum for a single photon is hv p  c Solar Sailing • Pathway to the Stars • Interplanetary travel without fuel • Gossamer technology

a form of using the radiation pressure (also called solar pressure) from stars to push large ultra-thin mirrors to high speeds. Unconventional Propulsion • Solar Sails – Light has momentum, p=hƒ, solar sails take advantage of this fact. • Differential Sail – Similar to solar sail, however this uses a theoretical coating on one side that absorbs energy more than the Solar Sail other side, assumes there exists in space a constant background radiation that is constantly impinging on all sides of the sail.

Differential sail Laser Cooling Cool, trap and manipulate atoms, ions, micro-particles using laser light

Optical Teezers Optical Teezers Optical Teezers

https://www.youtube.com/watch?v=ju6wENPtXu8