Lecture6 - Lecture 6 Notes, 95.658, Spring 2012,...

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Lecture 6 Notes, 95.658, Spring 2012, Electromagnetic Theory II Dr. Christopher S. Baird, UMass Lowell 1. Radiation Introduction - We have learned about the propagation of waves, now let us investigate how they are generated. - Physically speaking, whenever the electric field changes in time, it creates a magnetic field which changes in time, which creates an electric field which changes in time, etc. - This creates a chain reaction that then propagates away from the system and exists independent of it. - Every time the fields change in time, waves are radiated or propagated. - Any acceleration of an electric charge creates a changing electric field so that all accelerating charges radiate electromagnetic waves. - The only system that does not radiate is a system where all the charges are completely at rest with respect to some inertial frame. - In the real physical world, no charges are ever completely at rest. Every bit of matter has some temperature, even if very small, which means its atoms and electrons are constantly experiencing thermal motion. These moving charges radiate waves. - Therefore, all systems are constantly radiating, typically in the infrared wavelengths. - The thermal radiation tends to be random and is typically viewed as undesired noise in a system that must be overcome by a meaningful signal. - “Electrostatics” is an approximation where the radiation due to thermal motion is small enough to be considered negligible. - Unfortunately, the types of radiation and their sources are quite numerous and involved and we only have time in this course to cover the simplest forms of radiation. But this should give some understanding of the mechanisms involved. 2. General Solution for Radiation - For the purpose of understanding radiation, assume we have charges oscillating in free space. There are no boundaries or materials present. - Start with Maxwell's equations. The Maxwell-Ampere Law states that currents and changing electric fields give rise to curling magnetic fields: ∇× B = 0 J 1 c 2 E t Maxwell-Ampere Law in Free Space - Define potentials so that: B =∇× A and E =−∇ − A t and plug in: ∇×∇× A = 0 J 1 c 2 t −∇ − A t ∇∇⋅ A −∇ 2 A = 0 J 1 c 2 d t − 1 c 2 2 A t 2 - Because the fields are defined in terms of the derivatives of the potentials, there is some freedom in choosing potentials and we will always end up with the same fields. - Any specific choice of potentials is called a “gauge”.
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- The Maxwell equations are “gauge invariant”, meaning that they hold the same even if we switch from one gauge to the next. - Let us use the Lorentz gauge, defined by: ∇⋅ A =− 1 c 2 ∂ t - This causes two terms in the Maxwell-Ampere Law to cancel, leaving us with: 2 A 1 c 2 2 A t 2 =−μ 0 J Maxwell-Ampere Law in Free Space in the Lorentz Gauge - The Maxwell-Ampere Law in free space in the Lorentz gauge for the potential has reduced to
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Lecture6 - Lecture 6 Notes, 95.658, Spring 2012,...

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