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NASSP Self-study Review 0f Electrodynamics. Created by Dr G B Tupper [email protected] The following is intended to provide a review of classical electrodynamics at the 2 nd and 3 rd year physics level, i.e. up to chapter 9 of Griffiths book, preparatory to Honours.

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Nassp self study review 0f electrodynamics

NASSP Self-studyReview 0f Electrodynamics

Created by Dr G B Tupper

[email protected]


The following is intended to provide a review of classical electrodynamics at the 2nd and 3rd year physics level, i.e. up to chapter 9 of Griffiths book, preparatory to Honours.

You will notice break points with questions. Try your best to answer them before proceeding on – it is an important part of the process!


Basics
Basics

  • Maxwell’s equations:

  • Lorentz force:


Basics1
Basics

  • Mathematical tools:

    • Gauss’ Theorem

    • Stokes’ Theorem

    • Gradient Theorem

    • Green’s function


Basics2
Basics

  • Mathematical tools:

    • Second derivatives

    • Product rules

  • Potentials


Questions
Questions

  • Where is “charge conservation”?

  • Where is Coulomb’s “law”?

  • Where is Biot-Savart “law”?

  • What about Ohm’s “law”?


Work done on charge
Work done on charge

  • Power (Lorentz)

  • Now

  • So

  • Use Ampere-Maxwell


Conservation of energy
Conservation of energy

  • Identity

  • Use Faraday

  • So


Poynting s theorem
Poynting’s Theorem

  • Define

    • Mechanical energy density

    • Electromagnetic energy density

    • Poynting vector

  • EM fields carry energy


Questions1
Questions

  • Problem: an infinite line charge along z-axis moves with velocity :

    Determine


Waves in vacuum
Waves in vacuum

  • Maxwell’s equations:

  • Curl of Faraday:


Waves in vacuum1
Waves in vacuum

  • Use Gauss & Ampere-Maxwell; wave equation

  • Speed of light

  • Monochromatic plane-wave solutions

constant

Transverse


Questions2
Questions

  • What is the meaning of the wave-number ?

  • What is the meaning of angular frequency ?

  • What is the associated magnetic field?

Wavelength

Period



Monochromatic plane wave1
Monochromatic plane-wave

  • Energy density

  • Poynting vector

  • Momentum density


Monochromatic plane wave2
Monochromatic plane-wave

  • Time average

  • Intensity


Questions3
Questions

A monochromatic plane-polarized wave propagating in the z-direction has Cartesian components in phase:

.

In contrast, a circularly-polarized wave propagating in the z-direction has Cartesian components

  • out of phase:

  • Describe in words what such a circularly-polarized wave looks like. One of the two casess “left-handed”, and the other is “right handed” – which is which?

i

Determine the corresponding magnetic field.

Determine the instantaneous energy-density and Poynting vector.


Electro statics in matter
Electrostatics in matter

  • Electric field polarizes matter

    • Potential in dipole approximation

    • Bound charge density

Polarization: dipole moment

per unit volume


Electro statics in matter1
Electrostatics in matter

  • Rewrite Gauss’ law

    • Displacement field

    • For linear isotropic media

Free charge density



Magneto statics in matter
Magnetostaticsin matter

  • Magnetic field magnetizes matter

    • Vector potential

Magnetization: magnetic moment per

unit volume


Magneto statics in matter dipole moment proof
Magnetostaticsin matter;Dipole moment proof

  • Picture

  • Dipole approximation

  • For arbitrary constant vector

  • Therefore

=0

Q.E.D.


Magneto statics in matter1
Magnetostaticsin matter

  • Bound current density

  • Rewrite Ampere’s law

    • Induction

    • For linear isotropic media

  • Free current density


    Electro dynamics in matter
    Electrodynamics in matter

    • New feature

    • Rewrite Ampere-Maxwell


    Electro dynamics in matter1
    Electrodynamics in matter

    • Maxwell’s equations

    • Constitutive relations

    • Linear isotropic media


    Electro dynamics in matter2
    Electrodynamics in matter

    • Boundary conditions


    Electro dynamics in matter3
    Electrodynamics in matter

    • Energy density

    • Poynting vector


    Electromagnetic waves in matter
    Electromagnetic waves in matter

    • Assume electrical neutrality

    • In general there may be mobile charges; use

      • Resistivity

    Conductivity


    Electromagnetic waves in matter1
    Electromagnetic waves in matter

    • Maxwell’s equations

      • Curl of Faraday

      • For constant use Ampere-Maxwell


    Electromagnetic waves in matter2
    Electromagnetic waves in matter

    • Wave equation

    • In an ideal insulator

      • Phase velocity

      • Plane wave solution

    New

    Index of refraction


    Questions4
    Questions

    • What do you expect happens in real matter where the conductivity doesn’t vanish?

    • Which is more basic: wavelength or frequency?


    Electromagnetic waves in matter3
    Electromagnetic waves in matter

    • Take propagation along z-axis

      • Complex ‘ansatz’

      • Substitution gives

      • Solution


    Electromagnetic waves in matter4
    Electromagnetic waves in matter

    • Thus general solution is

    Transverse

    Phase

    Attenuation!

    Frequency dependant: dispersion


    Electromagnetic waves in matter5
    Electromagnetic waves in matter

    • Limiting cases

      • High frequency

      • Low frequency

    Good insulator

    Good conductor

    Note: at very high frequencies conductivity is frequency dependant


    Electromagnetic waves in matter6
    Electromagnetic waves in matter

    • Magnetic field – take for simplicity



    Questions5
    Questions

    What one calls a “good conductor” or “good insulator” is actually frequency dependant; i.e. is

    or ?

    Find the value of for pure water and for copper metal. Where does it lie in the electromagnetic spectrum in each case?

    For each determine the high-frequency skin depth.

    For each determine the skin depth of infrared radiation ( ).

    In the case of copper, what is the phase velocity of infrared radiation?

    In the case of copper, what is the ratio for infrared radiation?


    Frequency dependence
    Frequency dependence

    • Electric field polarizes matter

    • Model

    …dynamically

    Damping (radiation)

    “Restoring force”

    Driving force


    Frequency dependence1
    Frequency dependence

    • Electromagnetic wave

      • Rewrite in complex form

      • Steady state solution

    Natural frequency


    Frequency dependence2
    Frequency dependence

    • Substitution of steady state solution

    • Dipole moment


    Frequency dependence3
    Frequency dependence

    • Polarization

    • Complex permittivity

    Number of atoms/molecules per unit volume


    Frequency dependence4
    Frequency dependence

    • Even for a “good insulator”

    • Low density (gases)

    Absorption coefficient

    Ignore paramagnetism/diamagnetism


    Frequency dependence5
    Frequency dependence

    • Low density

    Frequency dependent: dispersion


    Frequency dependence6
    Frequency dependence

    Anomalous dispersion



    Electromagnetic waves in plasma
    Electromagnetic waves in Plasma

    • Electrons free to move; inertia keeps positive ions almost stationary

    • Model

      • Solution

    Electron mass

    No restoring force!


    Electromagnetic waves in plasma1
    Electromagnetic waves in Plasma

    • Current density

    • Conductivity

    Electron number density

    Drude model


    Electromagnetic waves in plasma2
    Electromagnetic waves in Plasma

    • Electron collisions rare, so dissipation small

    • Recall for conductor

    Purely imaginary!!


    Electromagnetic waves in plasma3
    Electromagnetic waves in Plasma

    • As

      • Above the plasma frequency: waves propagate with negligible loss

      • Below the plasma frequency: no propagation, only exponential damping

    Dispersion relation

    Plasma frequency

    F&F 2013 L46



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