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November 15, 2015 - February 4, 2021
Linear differential equations
Superposition of solutions
Finally, we make some remarks on why linear systems are so important. The answer is simple: because we can solve them! So most of the time we solve linear problems. Second (and most important), it turns out that the fundamental laws of physics are often linear.
A nonlinear equation cannot be solved, ordinarily, any other way but numerically.
Oscillations in linear systems
Analogs in physics
Series and parallel impedances
Optics: The Principle of Least Time
Light
Reflection and refraction
This, then, is one of the important steps in the development of physical law: first we observe an effect, then we measure it and list it in a table; then we try to find the rule by which one thing can be connected with another.
Fermat’s principle of least time
Fermat’s principle. His idea is this: that out of all possible paths that it might take to get from one point to another, light takes the path which requires the shortest time.
Thus the statement that the angle of incidence equals the angle of reflection is equivalent to the statement that the light goes to the mirror in such a way that it comes back to the point in the least possible time.
Applications of Fermat’s principle
When a new theoretical principle is developed, such as the principle of least time, our first inclination might be to say, “Well, that is very pretty; it is delightful; but the question is, does it help at all in understanding the physics?”
However, the importance of a powerful principle is that it predicts new things.
A more precise statement of Fermat’s principle
How it works
Geometrical Optics
Introduction
The focal length of a spherical surface
If the light really comes to a point, it is a real image. But if the light appears to be coming from a point, a fictitious point different from the original point, it is a virtual image.
The focal length of a lens
Magnification
Compound lenses
Any optical instrument—a telescope or a microscope with any number of lenses and mirrors—has the following property: There exist two planes, called the principal planes of the system (these planes are often fairly close to the first surface of the first lens and the last surface of the last lens), which have the following properties: (1) If light comes into the system parallel from the first side, it comes out at a certain focus, at a distance from the second principal plane equal to the focal length, just as though the system were a thin lens situated at this plane. (2) If parallel light
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Aberrations
Remember that the principle of least time is not an accurate formulation, unlike the principle of conservation of energy or the principle of conservation of momentum. The principle of least time is only an approximation,
Resolving power
The general rule for the resolution of any optical instrument is this: two different point sources can be resolved only if one source is focused at such a point that the times for the maximal rays from the other source to reach that point, as compared with its own true image point, differ by more than one period.
Electromagnetic Radiation
Electromagnetism
Maxwell could say, when he was finished with his discovery, “Let there be electricity and magnetism, and there is light!”
We have written these down only for the purpose of showing the beauty of nature or, in a way, the power of mathematics.
The problem of how to handle the part of this field which is generated by the very charge on which we want the field to act is not yet solved today.
Radiation
The dipole radiator
Interference
In any case, we have checked those features that are of the greatest importance for our later applications, and we shall come back to study some of the other properties of electromagnetic waves next year.
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Interference
Electromagnetic waves
Energy of radiation
Sinusoidal waves
Two dipole radiators
The mathematics of interference
Diffraction
The resultant amplitude due to equal oscillators
The diffraction grating
