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.
16. Transforming Maxwell equations-intro and 1st equation
Maxwell's four equations:
We would like to consider how the equations need to be changed so that they apply in a lab aboard a space ship moving at velocity, V, in the x direction. The most obvious effect is that a stationary charge is now seen as moving in the negative x direction and thus being seen as a current density as well as a charge density.
We are trying to verify that the Lorentz transformations result in the same set of Maxwell's equations. Thus we will use the Lorentz transformations, as well as the transformed electric and magnetic field equations which resulted from the Lorentz transforms.
Transformation of partial derivatives
One tricky aspect of transforming Maxwell's equations from the stationary reference frame to the moving one is the need to transform the partial derivatives in those equations from using x, y, z, and t as variables into using x', y', z', and t'. In Fig. 16.1 we show the standard inverse Lorentz transform of these variables and some useful partial derivatives of these. We don't show the relationships of y and z to y' and z' since these are trivial, i.e. y' = y and z' = z and ∂y'/∂x=0, ∂y'/∂y=1, ∂y'/∂z=0, ∂y'/∂t=0, ∂z'/∂x=0, ∂z'/∂y=0, ∂z'/∂z=1, and ∂z'/∂t=0.
Fig. 16.1a. Inverse Lorentz transforms of x and t.
Fig. 16.1b. Partial derivatives of the Lorentz transforms at the left.
Fig. 16.1c. More partial derivatives of the Lorentz transforms at the left.
In Fig. 16.2 below we show the chain rule for partial derivatives applied to a typical electromagnetic field quantity, Ex in this example. We calculate the differential of Ex in terms of differential changes of the new variables x', y', z' and t'. This equation involves taking partial derivatives of Ex with respect to each of these new variables while the other new variable is held constant. One important aspect to note is which variables are held constant in the various terms.
Fig. 16.2. Calculation of the differential of a typical electromagnetic field quantity Ex in terms of differentials of the variables x' and y' of the moving reference frame.
In the next step, in Fig. 16.3, we form the partial derivative of Ex with respect to x, in the form it is found in the original Maxwell's equations, but now in terms of the partial derivatives in Fig. 16.2 above. This is one step in transforming Maxwell's equations from being written in terms of x, y, z, and t to being written in terms of x', y', z', and t'.
Fig. 16.3. Calculation of the partial derivative with respect to x of a typical field quantity (Ex shown here as an example). This operation is one step in changing the variables of Ex from x, y, z, and t into x', y', z', and t', i.e. into the moving reference frame. The second equation above (boxed) is simply the first one rewritten separating out the operations from Ex.
In Fig. 16.3, we have dropped the terms that involve partial derivatives with respect to y' and z' because:
The whole equation (see the left hand side) is directed towards calculating the partial of Ex with respect to x while keeping y, z, and t fixed.
But y' = y and z' = z.
Thus y' and z' are to be kept fixed and are not variables for this equation.
Thus we are only interested in the x' and t' dependence for this equation.
Fig. 16.4. Calculation of the partial with respect to t.
Transformation of the first Maxwell equation
We start out transforming the left side of the first Maxwell Equation (as shown in (16.1) above):
We next write out the transformation of the right side of the first Maxwell equation (see (16.1) above).
Finally, we use these in the first Maxwell equation (written with all terms moved to the left):
Fig. 16.5. First Maxwell equation transformed to moving reference frame.
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.