electrodynamics notes pdf

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Electrodynamics

This set of “lecture notes” is designed to support my personal teaching ac-tivities at Duke University in particular teaching its Physics 318/319 series (graduate level Classical Electrodynamics) using J D Jackson’s Classical Elec-trodynamics as a primary text However the notes may be useful to students

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INTRODUCTION TO ELECTRODYNAMICS

8 2 1 Newton’s Third Law in Electrodynamics 360 8 2 2 Maxwell’s Stress Tensor 362 8 2 3 Conservation of Momentum 366 8 2 4 Angular Momentum 370 8 3 Magnetic Forces Do No Work 373 9 Electromagnetic Waves 382 9 1 Waves in One Dimension 382 9 1 1 The Wave Equation 382 9 1 2 Sinusoidal Waves 385 9 1 3 Boundary Conditions: Reﬂection and

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LECTURE NOTES ON ELECTRODYNAMICS PHY 841 – 2017

Electromagnetism is inherently relativistic To see this consider a charged particle moving through a magnetic field in deep space The particle undergoes an acceleration proportional to its velocity because magnetic force F ⃗ = q⃗v R ⃗ depend on velocity But who defines the velocity? If an observer moves with the particle’s velocity the spee

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Chapter 7 Electrodynamics

Chapter 7 Electrodynamics 7 1 Electromotive Force An electric current is flowing when the electric charges are in motion In order to sustain an electric current we have to apply a force on these charges In most materials the current density is proportional to the force per unit charge: =s f

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Classical Electrodynamics

This set of “lecture notes” is designed to support my personal teach-ing activities at Duke University in particular teaching its Physics 318/319 series (graduate level Classical Electrodynamics) using J D Jackson’s Clas-sical Electrodynamics as a primary text However the notes may be useful

What topics are covered in electrodynamics?
A very brief review of the electrodynamics topics covered includes: plane waves, dispersion, penetration of waves at a boundary (skin depth), wave guides and cavities and the various (TE, TM, TEM) modes associated with them, radiation in the more general case beginning with sources.
What is classical electrodynamics?
Classical Electrodynamics is one of the most beautiful things in the world. Four simple vector equations (or one tensor equation and an asssociated dual) describe the unified electromagnetic field and more or less directly imply the theory of relativity.
Are these typeset notes based on the course 'electrodynamics'?
These notes are based on the course “Electrodynamics” given by Dr. M. J. Perry in Cambridge in the Michælmas Term 1997. These typeset notes have been prod uced mainly for my own benefit but seem to be officially supported. The recommended books for this course are discussed in the bibliography.
What is the fundamental problem electrodynamics hopes to solve?
The fundamental problem electrodynamics hopes to solve is this (Fig. 2.1): We have some electric charges,q 1,q 2,q 3,... (call themsource charges); what force do they exert on another charge,Q(call it thetest charge)? The positions of the source charges aregiven(as functions of time); the trajectory of the test particle isto be calculated.

Special Relativity Primer

Electromagnetism is inherently relativistic. To see this, consider a charged particle moving through a magnetic field in deep space. The particle undergoes an acceleration proportional to its velocity because magnetic force, F ⃗ = q⃗v R, ⃗ depend on velocity. But, who defines the velocity? If an observer moves with the particle’s velocity, the spee

2 Dynamics of Relativistic Point Particles

We won’t discuss the dynamics of individual charged particles very much in this course, but it is good to review the least-action principle, and the example for relativistic particles. In the next chapter we will apply these principles to field equations, deriving Maxwell’s equations, so it will be helpful to review the connection between least act

2.2 Interaction of a Charged Particle with an External Electromagnetic Field

Here, the external electromagnetic field is a four-vector Aμ(r). The zeroth component is the electric potential φ and the spatial components are the usual vector potential. To make a Lorentz-invariant action that has a contribution that looks like the usual potential energy, eφ(r), we consider S = Z dτ ( m eu A) − − · Z = m − s dt dt′ dt people.nscl.msu.edu

2.5 The Electromagnetic Field Tensor

One can define a second-rank anti-symmetric tensor using the vector potential, αβ = F ∂αAβ ∂βAα. − Using the definitions, people.nscl.msu.edu

3 Dynamic Electromagnetic Fields

So far we have discussed the motion of particles in a field, but have ignored how the fields might change in time. To do so, we need to consider all three parts of the action: the action of a free particle, Sm, the action involving the interaction of matter with the field Sfm and, new for this chapter, the action of the field, Sf. We will show how

3.2 Pseudo-Vectors and Pseudo-Scalars

The vector potential Aα is a four-vector, and the field tensor F αβ is a second-rank tensor. The electric field E ⃗ = A0 + ∂t A ⃗ is a 3-vector, but one that transforms as part of a tensor if the −∇ transformation involves a boost. The magnetic field B ⃗ = A ⃗ is a pseudo-vector. The "pseudo" comes from the fact that its definition, εijk∂jAk, invol

2( E ⃗ B ⃗ 2).

− − (3.23) Thus, although E ⃗ and B ⃗ mix under boosts, the difference of their magnitudes remains fixed. The sign of a pseudo-vector or pseudo-scalar changes if one changes from a right-handed to a left-handed coordinate system. This is because εijk was arbitrarily defined so that εxyz was positive. Even though magnetic forces feature pseud

3.4 Hyper-Surfaces and Conservation of Energy, Momentum and Angular Momentum

The energy and momentum, P α, in a three-dimensional hyper-surface element Ωγ is Ω P α = people.nscl.msu.edu

4 Electrostatics

Here, we consider the electric field of fixed charge distributions. All currents are set to zero, so there is only electric field, and all time derivatives in Maxwell’s equations are neglected. people.nscl.msu.edu

Example 4.1:

Find the net potential energy for a charge Q uniformly spread out in a sphere of radius R. First, find the potential Φ(r). For r > R, it is easy, Φ = Q/r. For r < R, you need to first find the electric field. Beginning with Gauss’s law, people.nscl.msu.edu

4.4 Laplace’s Equations in Cartesian Coordinates

In Cartesian coordinates separation of variables assumes that Φ is a product of three pieces, people.nscl.msu.edu

4.7 Boundary Value Problems

Boundary value problems involve finding solutions for Laplace’s equations that satisfy the B.C. for some region of space. The B.C. must be satisfied at all boundaries of the space. Often, the boundaries are either a conductor, constant potential, or at infinity, with the potential either vanishing or behaving with a known manner, e.g. becoming a co

Example 4.2:

Consider a point charge +Q outside a grounded conducting sphere. The sphere has radius R and is centered at the origin, and the point charge is at position aˆ z. people.nscl.msu.edu

(4.52) 4.7.3 Boundary Value Problems Using Spherical Harmonics

Laplace’s equation is applicable in any charge-free region, but doesn’t mean it doesn’t apply in problems with charge densities. You simply only use Laplace’s equation in the charge-free part of the volume. There are a few, with emphasis on few, boundary-value problems one can easily perform using the spherical harmonics mentioned before. The most

4.7.4 Boundary Value Problems Using Cylindrical Harmonics

This is very similar in spirit to the spherical case. people.nscl.msu.edu

5 Multipole Expansions

Here, we consider fields due to compact figurations of static charges when viewed from far away. The fields are dominated by the lowest non-zero moment of the charge distributions, e.g. monopole, dipole, quadrupole, etc. people.nscl.msu.edu

p T

= . tanh(pE/T ) − E For high temperatures, one can expand in pE/T and find pz⟩ ⟨ people.nscl.msu.edu

5.3 Energy in arbitrary external field

As stated earlier, the energy of a charge in an external potential Φ, or that of a dipole in an external electric field is U(monopole) = people.nscl.msu.edu

6 Magnetostatics

Here, we consider magnetic fields from steady currents, and set the stage for fields from dynamic sources. people.nscl.msu.edu

6.4 Overlapping Distributions and Hyperfine Splitting

Thus far, we have considered only the interaction between to distributions separated by a finite amount. Sometimes, e.g. electrons interacting with a nucleus to produce the hyper-fine inter-action, the two distributions lay atop one another. Here, we consider the electric and magnetic field integrated over volumes that include all charges. This doe

7 Electromagnetic Waves

In the last several sections we have considered static systems, where we could neglect all the ∂t terms in Maxwell’s equations. In this chapter we consider the propagation of waves, and · · · wave equations clearly require the ∂t

7.4 Wave Guides and Cavities

Here we consider oscillating solutions at fixed frequency. We consider the simplest case, that of a vacuum surrounded by a perfect conductor. For non-perfect conductors the boundary condi-tions are more complicated due to the penetration of the fields into the media. Such penetration also leads to damping of waves. Because this course explicitly ig

= qq2

+ q2 y. The procedure can be followed for any cross-sectional shape, assuming the wave-guide is trans-lationally invariant along the z axis. One can always divide the solutions into TM and TE modes. For the TM modes, one can solve the boundary conditions first for the function ψ(x, y), which gives Ez(x, y) by the relation, people.nscl.msu.edu

8 Radiation

Here we discuss classical radiation. The source of such radiation is the J A terms in the La- · grangian, and the “classical” assumption is that the current does not change due to the radiated photon. In contrast, quantum emission involves changing discrete levels. For instance one could fall from a p-state to an s-state, with the frequency determi

1 ˆn ⃗v

− · where ˆn is the unit vector parallel to ⃗x ⃗r. As was seen in the last section, in the non-relativistic − limit this factor simply provides the inverse distance from the point of emission to the observer. However, relativistically the additional factor (1 ˆn ⃗v)− 1 amplifies the response of the potential − · to the charge when the velocity appr

E(ω) ⃗

in terms of an integral involving only the ̃ E(ω) ⃗ t′=tf people.nscl.msu.edu

9 Scattering

If the object from which light is scattered is small compared to the wavelength of light, it is convenient to think of the problem in two steps. First, the object is excited by the electromagnetic wave, inducing oscillating multipole moments of either the current or charge. These objects then radiate according to the moments. The incoming electroma

Introduction (Introduction to Electrodynamics)

L1.1 The Realms of Mechanics Introduction to Electrodynamics D.J. Griffiths

L2.1 The Natural Forces Introduction to Electrodynamics D.J. Griffiths

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