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NASSP Self-study Review 0f Electrodynamics Created by Dr G B Tupper

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

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Basics Maxwells equations: Lorentz force:

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Basics Mathematical tools: – Gauss Theorem – Stokes Theorem – Gradient Theorem – Greens function

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Basics Mathematical tools: – Second derivatives – Product rules Potentials

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Questions Where is charge conservation? Where is Coulombs law? Where is Biot-Savart law? What about Ohms law? Where is charge conservation? Where is Coulombs law? Where is Biot-Savart law? What about Ohms law?

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Work done on charge Power (Lorentz) Now So Use Ampere-Maxwell

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Conservation of energy Identity Use Faraday So

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Poyntings Theorem Define – Mechanical energy density – Electromagnetic energy density – Poynting vector EM fields carry energy

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

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Waves in vacuum Maxwells equations: Curl of Faraday:

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Waves in vacuum Use Gauss & Ampere-Maxwell; wave equation Speed of light Monochromatic plane-wave solutions constant Transverse

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Questions What is the meaning of the wave-number ? What is the meaning of angular frequency ? What is the associated magnetic field? Wavelength Period

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Monochromatic plane-wave

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Energy density Poynting vector Momentum density

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Monochromatic plane-wave Time average Intensity

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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.

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Electrostatics in matter Electric field polarizes matter – Potential in dipole approximation – Bound charge density Polarization: dipole moment per unit volume Polarization: dipole moment per unit volume

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Electrostatics in matter Rewrite Gauss law – Displacement field – For linear isotropic media Free charge density

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Dielectric constant

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Magnetostatics in matter Magnetic field magnetizes matter – Vector potential Magnetization: magnetic moment per unit volume Magnetization: magnetic moment per unit volume

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Magnetostatics in matter; Dipole moment proof Picture Dipole approximation For arbitrary constant vector Therefore =0 Q.E.D.

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Magnetostatics in matter – Bound current density Rewrite Amperes law – Induction – For linear isotropic media Free current density

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Electrodynamics in matter New feature Rewrite Ampere-Maxwell

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Electrodynamics in matter Maxwells equations Constitutive relations Linear isotropic media

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Electrodynamics in matter Boundary conditions

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Electrodynamics in matter Energy density Poynting vector

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Electromagnetic waves in matter Assume electrical neutrality In general there may be mobile charges; use – Resistivity Conductivity

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Electromagnetic waves in matter Maxwells equations – Curl of Faraday – For constant use Ampere-Maxwell

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Electromagnetic waves in matter Wave equation In an ideal insulator – Phase velocity – Plane wave solution New Index of refraction

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Questions 1.What do you expect happens in real matter where the conductivity doesnt vanish? 2.Which is more basic: wavelength or frequency?

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Electromagnetic waves in matter Take propagation along z-axis – Complex ansatz – Substitution gives – Solution

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Electromagnetic waves in matter Thus general solution is Transverse Phase Attenuation! Frequency dependant: dispersion

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Electromagnetic waves in matter Limiting cases – High frequency – Low frequency Good insulator Good conductor Note: at very high frequencies conductivity is frequency dependant

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Electromagnetic waves in matter Magnetic field – take for simplicity

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Electromagnetic waves in matter Good conductor

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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?

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Frequency dependence Electric field polarizes matter Model Restoring force Driving force Damping (radiation) …dynamically

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Frequency dependence Electromagnetic wave – Rewrite in complex form – Steady state solution Natural frequency

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Frequency dependence – Substitution of steady state solution – Dipole moment

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Frequency dependence Polarization Complex permittivity Number of atoms/molecules per unit volume

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Frequency dependence Even for a good insulator Low density (gases) Ignore paramagnetism/diamagnetism Absorption coefficient

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Frequency dependence Low density Frequency dependent: dispersion

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Frequency dependence Anomalous dispersion

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Questions

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Electromagnetic waves in Plasma Electrons free to move; inertia keeps positive ions almost stationary Model – Solution Electron mass No restoring force!

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Electromagnetic waves in Plasma Current density Conductivity Electron number density Drude model

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Electromagnetic waves in Plasma Electron collisions rare, so dissipation small Recall for conductor Purely imaginary!!

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Electromagnetic waves in Plasma As – Above the plasma frequency: waves propagate with negligible loss – Below the plasma frequency: no propagation, only exponential damping Dispersion relationPlasma frequency F&F 2013 L46

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Plasma - Ionosphere

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