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Combinations of lenses, geometric optics

Equations:

The thin lens equation relates the distances to the image and object, d_i and d_o to the focal length f according to

The distance to the object and image are positive if they are real, and negative if they are virtual. The focal length is positive for converging lenses and negative for diverging lenses. The magnification of a lens is the ratio of the height h_i of the image to the height h_o of the object, and is given by

Cameras, the eye, magnifiers, telescopes, microscopes, aberration, diffraction limits.

Equations:

Cameras:

Eyes: Nearsightedness can be corrected by a diverging lens whose focal point is at the furthest point where the eye can naturally focus when relaxed. Farsightedness can be corrected with a converging lens which put the image of an object 25 cm away at the location of the eye's actual near point, which is further away. The thin lens equation is used in either case to find the focal length needed.

Magnifier: The magnification with the image at infinity is

In a telescope, parallel rays from a distant object are focused at the focal point of the objective lens, which is in turn the focal point of the eyepiece. Parallel rays then go to the eye, which sees the object at infinity.

Telescope with objective focal length f_o and eyepiece focal length f_i has magnification:

In microscopes, the object is placed just beyond the focal point of the objective lens, producing an enlarged image at the focal point of the eyepiece, which acts as a magnifier to produce an image at infinity. The magnification of the objective lens is

The size of a lens aperture limits the resolution due to diffraction through the opening, which creates a peak whose half-width (width to first minimum) is

If a material of index of refraction n fills the space below the objective lens, the resolving power is smaller by a factor of 1/n, because the wavelength is increased by a factor of n compared to in a vacuum.

Units:

1 Diopter = 1 m^-1 in inverse focal length

The principle of relativity (physics is the same in all intertial frames), constancy of the speed of light in a vacuum, simultaneity, time dilation, length contraction, mass increase, relations between energy and momentum, addition of velocities.

Equations:

Let g = (1-v²/c²)^-1/2. This factor is always greater than 1, and represents the time dilation factor by which events happen more slowly in a moving reference frame. A person taking a trip on a fast rocket will age more slowly by this factor. The mass of a moving object will increase by the same factor, and its length will decrease by this factor, becoming L / g.

The relativistic momentum is

The kinetic energy of a moving object is

The total relativistic energy satisfies

The relativistic sum of parallel velocities u and v is

Convenient units for masses of relativistic particles are eV/c². Convenient units of momentum are eV/c. For higher energies, substitute keV, MeV, ... as needed.

Discovery of the electron, blackbody radiation and Planck's quantum hypothesis, the photoelectric effect and photons, photon interactions, wave-particle duality, the wave nature of matter (de Broglie waves), electron microscopes, early models of the atom, atomic spectra, the Bohr atom.

Equations:

There are many equations in this chapter. The following list contains some central ones, but is not complete. Please see the chapter for a complete list.

Planck's quantum hypothesis says that molecular or atomic vibrations have a minimum energy related to their frequency f by E_min = hf with Planck's Constant h = 6.626 x 10^-34Js.

Photons are the basic particle constituents of electromagnetic fields. they carry energy

In the photoelectric effect, if is initially bound with a potential energy -W, which requires work W to overcome, then it is ejected from an atom with kinetic energy

de Broglie suggested that the relation between momentum and wavelength of photons applies to all objects: they have a wavelength related to their momentum by

Electron microscopes use the short wavelength of electrons to probe smaller objects than optical microscopes can.

The spectral lines of hydrogen have wavelengths given by the Rydberg law

The spectral lines of single-electron atoms can be found using Bohar's quantum condition

The quantized energy levels are given by

The spectral lines occur when the photon has an energy which is the difference between two of these quantized energy levels: