CS 4594 Broadband Electricity and Magnetism

Applications Applications of electricity and magnetism –Light –Magnetism –Motors –Radio –Xrays –Chemistry – molecular bonds

Electricity and Magnetism Electricity and Magnetism are part of the same physical phenomenon. Together they form a six-dimensional force field. This electricity and magnetism is also represented by –a four component potential field and by –electric currents and charge distributions

Maxwell’s Equations Maxwell brought together the works of his predecessors (Faraday, Coulomb, Gauss, Ampere, Biot and Savart, etc.) who were studying experiments with electric currents and magnets.

Main Results Matter can carry both positive and negative electric charge. Moving charge forms electric currents in such a way that the total charge is conserved. Electrostatic forces attract and repel electrically charged matter at distance (inverse square law). Similarly charged mater repels each other and oppositely charged matter attracts each other. Matter can be magnetized, but magnetic poles occur in positive/negative pairs. Magnetic forces attract and repel magnetized matter. Like poles repel and oppositely magnetized poles repel. Changing electric currents generate magnetic forces.

Maxwell’s Equations (div and curl)

Constitutive Equations The constitutive equations related the EB with the DH fields In a vacuum P and M are zero. The  0 and  0 are constants in a vacuum. They vary according to the media. Most media such as air and glass have electric currents and fields that are a part of their structure. This is represented by P and M. In many media including air and glass P is approximately proportional to E and M is approximately proportional to B.

Dialectric Constants Dialectric ε –Vacuum: 1 –Air: –Glass: 4 to 7 Magnetic Permeability –Vacuum 1 –non-magnet materials: close to 1

Lorenz Force The E,B field acts linearly on moving charge (electric current) according to the Lorenz formula The E,B field can be detected by moving a test charge and seeing the force on it Here,  is the charge and v the velocity.

Potential The first half of Maxwell’s equations imply that there is a 4D potential field. This is related to conservation of energy and momentum.

Conservation of Charge The second half of Maxwell’s equations say that there is conservation of charge.

Modern Version Maxwell’s equations show that time and space are related in what is known as 4D space-time. (1 time + 3 space) In this 4D world integration and differentials, are defined using an algebra of differential forms. Maxwell’s equations become: In addition because of dF=0: dA = F

Wave equation From these equations comes a set of second order linear differential equation called the wave equation. In holds in vacuum and media such as air and glass. Here c is the speed of light (related to  and  by the formula the c=sqrt(  )

Solutions of Wave Equation Solutions of the wave equation can be decomposed into vector functions involving the trigonometric functions sine and cosine. This holds in vacuum and media such as air and glass. When electromagnetic radiation hits conductive material such as metal and water. The internal arrangement of charge interacts with the electromagnetic fields to produce complicated effects such as reflection, scattering, and absorption.

Characteristics of Radio Waves Radio waves –travel at the speed of light. This follows from the wave equation. –can vibrate at a wide range of frequencies, including a wide variety of phenomena: microwave, infrared, light, UV, Xray, and gamma rays. –can be reflected as they bounce off material and refracted as they traverse different media

Wave Length If is the wave length and f is the frequency, then =c/f, where c is the speed of light. c = 3  10 8 meters/second.

Electromagnetic Spectrum

Reflection and Refraction Incident ray n 1 =c/v 1 n 2 =c/v 2 Reflected ray Refracted ray θfθf θiθi θrθr

Snell’s Law For refraction, (Here θ 1 = θ i the angle of incidence and θ 2 = θ f the angle of refraction.)

Snell’s Law Incident ray n 1 =c/v 1 n 2 =c/v 2 Refracted ray θ2θ2 θ1θ1