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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 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!

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Basics Maxwell’s equations: Lorentz force:

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**Basics Mathematical tools: Gauss’ Theorem Stokes’ Theorem**

Gradient Theorem Green’s function

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Basics Mathematical tools: Second derivatives Product rules Potentials

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**Questions Where is “charge conservation”? Where is Coulomb’s “law”?**

Where is Biot-Savart “law”? What about Ohm’s “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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**Poynting’s Theorem Define EM fields carry energy**

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 Maxwell’s 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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**Monochromatic plane-wave**

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

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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

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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 Ampere’s 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**

Maxwell’s 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**

Maxwell’s 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 What do you expect happens in real matter where the conductivity doesn’t vanish? 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 …dynamically**

Model …dynamically Damping (radiation) “Restoring force” Driving force

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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)**

Absorption coefficient Ignore paramagnetism/diamagnetism

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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**

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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Plasma - Ionosphere

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Maxwell’s Equations in Vacuum (1) .E = / o Poisson’s Equation (2) .B = 0No magnetic monopoles (3) x E = -∂B/∂t Faraday’s Law (4) x B = o j.

Maxwell’s Equations in Vacuum (1) .E = / o Poisson’s Equation (2) .B = 0No magnetic monopoles (3) x E = -∂B/∂t Faraday’s Law (4) x B = o j.

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