Their applications, physics, and math. -- Peter Ceperley
There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.
15. Transformation of electric and magnetic fields
We cannot feel or see electric and magnetic fields. We hypothesize them to account for forces and trajectories of charged particles and currents. Electric and magnetic fields are detected or "felt" with charges and currents. Along these lines, in order to figure out how electric and magnetic fields transform under change in reference frame, we must see how the Lorentz force changes upon change of reference frames:
Box 15.1. Writing the force F in terms of forces in the primed reference frame
To start, we need the transformation for force. We define force by the effect it has on the motion of objects using Newton's second law:
F = dp/dt . (15.2)
We need the momentum transform from Chapter 11, which to repeat is:
where ε is energy as discussed in Chapter 11. Script "E" is used to distinguish energy from electric field. We've also assumed that the velocity V of the reference frame does not change, which also means γ does not change either.
Dividing dp by dt gives us the force in Equation (15.2):
Dividing both the numerator and denominator by dt' and replacing the components of dp'/dt' with the corresponding components of F', we have:
We can relate the time derivative of energy in the above equation to force using:
This makes (15.5) become:
Continuing to simply:
Next we need to transform the components of u' into corresponding components of u. This may seem like a step backwards, but in the end we need primed variables times unprimed velocities, u, to match the form of (15.1) above where E and B will be functions of E', B' and V (the relative velocity of the primed and unprimed reference frames).
Box 15.2. Converting the velocities in the primed reference frame to the unprimed frame.
We start by repeating the transform for velocity from Chapter 9:
The messiest factor in this transform will be converting the u'x in the 1 + u'xV/c2 factor occurring in the denominators of (15.8) to ux. We first transform and simplify this factor:
We use this to transform the u' in (15.8) into corresponding components of u as promised:
We cancel factors as appropriate to write:
Box 15.3. Introducing electric and magnetic fields.
We next need to substitute the Lorentz force (containing electric and magnetic fields) in for the forces. Just to remind you, the Lorentz force in the primed reference frame is:
When we substitute the various force components of (15.11) into (15.10), F becomes:
This last substitution has reintroduced some primed velocities. We transform these into velocities in the unprimed reference frame using (15.9):
With some rearrangement, this becomes
where a number of terms in the first line have been collected and designated as T given by:
T can be further simplified by the following steps:
Substituting this into (15.14) we have:
We need to interpret the above in terms of the standard Lorentz force:
Matching up (15.16) with (15.17), looking at the multipliers of the various velocities, we see that (15.16) will become (15.17) provided that
Below we have repeated (15.18) as well as a vector notation of the results (in terms of the parallel and perpendicular components). These are the transforms of the electric and magnetic fields to be used when changing from one reference frame to another.
Waves, Berkeley Physics Course - vol. 3, Frank S. Crawford, Jr. McGraw-Hill 1965. This book is suitable for an add-on to an introductory course on college or university physics. It discusses all sorts of aspects of waves and has a multitude of home experiments. One could probably make a great science fair project from one of them. As to its math level, it mostly uses algebra, with some calculus in the mix.
Physics of waves, by Elmore and Heald, originally published by McGraw-Hill in 1969, but currently published by Dover. This book covers many different wave systems, such as waves on a string, on a membrane, in solids, in fluids, on a liquid surface, and electromagnetic waves. It also covers the many aspects of waves. It has an excellent chapter on diffraction.
The Feynman lectures on physics, Feynman, Leighton, and Sands, Addison-Wesley 1963. Three volumes. These cover many aspects of physics. They are perhaps best suited for someone who has made it through an introductory sequence in college or university physics, and wants to read about the subject from a more sophisticated point of view. They are not particularly math intensive, more just into discussing concepts with some math as required. These are books you read to understand a physicist's mind. Perhaps 10% to 20% of the chapters are about waves and resonances.
Electromagnetic books that I use:
Engineering Electromagnetics, Hayt (with Buck on more recent editions), McGraw-Hill. An easy to read, compact junior-level text for electrical engineering students.
Fields and waves in communication electronics, Ramo, Whinnery, Van Duzer, Wiley. A upper level/graduate level text for electrical engineering student. Covers practically every aspect of applied electromagnetic fields in some depth. Is not a book to sit down and read for philosophy, but rather to look up the rational behind certain devices or design methods.