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Physics

Ray Optics and Optical Instruments

Optical Fibers

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⚡ Quick Summary
Optical fibers use total internal reflection to transmit light signals over long distances with minimal loss. They are used in light pipes for seeing hard-to-reach places and for transmitting communication signals.
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Optical fibers are thin strands of plastic with a radius on the order of a micrometer. A bundle of such fibers forms a light pipe. Light entering the fiber makes a glancing incidence on the wall, exceeding the critical angle, causing total internal reflection and guiding the light along the fiber. Applications include: * **Light Pipes:** Viewing inaccessible areas (e.g., inside the human body). * **Communication Signals:** Transmitting signals by mixing them with light waves, offering better clarity than conventional methods. Optical fibers are typically coated with a material having a lower refractive index than the fiber to ensure optical insulation.

Prism and Deviation

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A prism refracts light, causing it to deviate from its original path. The angle of deviation depends on the angle of incidence, the prism's angle, and the refractive indices of the prism and surrounding medium. The angle of minimum deviation occurs when the light passes symmetrically through the prism.
["d = (i + i') - (r + r')", "r + r' = A", "d = i + i' - A"]
A prism consists of refracting surfaces AB and AC, with angle BAC being the prism angle (A). A ray PQ incident on surface AB is refracted along QR. The angles of incidence and refraction are denoted as i and r respectively. The ray QR then strikes surface AC. If the incidence angle (r') exceeds the critical angle, total internal reflection may occur; otherwise, the ray is refracted along RS, exiting with an angle of refraction i' (also called the angle of emergence). The angle of deviation (δ) represents the difference between the incident ray's original path and the final refracted ray. It can be calculated as: d = (i - r) + (i' - r') = (i + i') - (r + r') From the geometry of the prism: r + r' = A Therefore, the angle of deviation can be expressed as: d = i + i' - A

Angle of Minimum Deviation

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The minimum deviation occurs when the incident ray passes symmetrically through the prism (i = i').
i = i' at minimum deviation
The angle of deviation 'd' is determined by the angle of incidence 'i'. For a specific 'i', the deviation is minimum. At minimum deviation, the ray passes symmetrically, so i = i'.

Combination of Lenses

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The equivalent focal length and position of a combination of lenses can be calculated to determine the behavior of light passing through the system.
D = dF/f where D is the distance behind the second lens, d is the distance, F is the focal length, and f is the focal length of the second lens. Real depth/Apparent depth = μ (refractive index)
When lenses are combined, their individual focal lengths and the distances between them determine the behavior of the combined system. Calculations involving the positions of lenses and the focal lengths are essential for determining the location and nature of the final image.

Refraction at Spherical Surface

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Light bends when it enters a curved surface, and we can predict how using a formula that connects refractive indices, distances, and curvature.
(μ₂/v) - (μ₁/u) = (μ₂ - μ₁)/R where μ₁ and μ₂ are refractive indices, v and u are image and object distances, and R is the radius of curvature.
The refraction at a spherical surface depends on the refractive indices of the two media, the object and image distances, and the radius of curvature of the surface. The formula describing this relationship is fundamental in understanding how lenses form images.

Reflection from Plane Mirrors

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Plane mirrors reflect light in a predictable way, forming virtual images that appear to be behind the mirror.
Real depth / Apparent depth = μ₂/μ₁
The nature of images formed by plane mirrors can be understood in terms of how reflected rays appear to originate from a point behind the mirror. All reflected rays meet at a point when produced backward.

Lens Formula

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Relates object distance (u), image distance (v), and focal length (f) of a lens.
1/v - 1/u = 1/f
The lens formula is given by 1/v - 1/u = 1/f, where: - v is the image distance - u is the object distance - f is the focal length of the lens

Linear Magnification

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Ratio of image height to object height, also related to image and object distances.
m = v/u
Linear magnification (m) is defined as the ratio of the height of the image (h') to the height of the object (h). It can also be expressed as the negative ratio of image distance (v) to object distance (u): m = h'/h = -v/u

Angular Magnification of Compound Microscope

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The magnification when the final image is formed at the least distance of distinct vision.
m = (v_o / u_o) * (1 + D/f_e)
When the final image is formed at the least distance of distinct vision (D), the angular magnification (m) of a compound microscope is approximately given by: m = (v_o / u_o) * (1 + D/f_e) Where: - v_o is the image distance for the objective lens - u_o is the object distance for the objective lens - D is the least distance of distinct vision (typically 25 cm) - f_e is the focal length of the eyepiece

Angular Magnification of Compound Microscope (Final Image at Infinity)

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The angular magnification when the final image is formed at infinity.
m = v_o / (u_o * f_e)
When the final image is formed at infinity, the angular magnification (m) of a compound microscope is approximately given by: m = v_o / (u_o * f_e) Where: - v_o is the image distance for the objective lens - u_o is the object distance for the objective lens - f_e is the focal length of the eyepiece

Dispersion through a Prism Combination

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This section discusses how to combine two prisms with different refractive indices and apex angles to achieve either dispersion without average deviation or average deviation without dispersion. Formulas are derived for the net deviation and angular dispersion of such combinations.
["d = (μ - 1)A - (μ' - 1)A'", "δ = (μv - μr)A - (μ'v - μ'r)A'", 'ω = (μv - μr) / (μy - 1)', "δ = (μy - 1)ωA - (μ'y - 1)ω'A'", "(μy - 1)A = (μ'y - 1)A' (for dispersion without average deviation)", "δ = (μy - 1)A (ω - ω') (for dispersion without average deviation)", "(μy - 1)ωA = (μ'y - 1)ω'A' (for average deviation without dispersion)", "(μv - μr)A = (μ'v - μ'r)A' (for average deviation without dispersion)", "d = (μy - 1)A [1 - ((μ'y - 1)A') / ((μy - 1)A)] (for average deviation without dispersion)"]

Dispersion through a Prism Combination

When two prisms of different materials and apex angles are combined, the net deviation and dispersion depend on the properties of each prism.

Net Deviation

The net deviation (d) produced by the combination of two prisms is the difference between the deviations produced by each prism individually.

d = d1 - d2 = (μ - 1)A - (μ' - 1)A'

Angular Dispersion

The angular dispersion (δ) produced by the combination is the difference in deviation between the violet and red rays.

δ = dv - dr = (μv - μr)A - (μ'v - μ'r)A'

Dispersive Power

Dispersive power (ω) is defined as:

ω = (μv - μr) / (μy - 1)

Where μv, μr, and μy are the refractive indices for violet, red, and yellow light, respectively.

Therefore, the net angular dispersion can be written as:

δ = (μy - 1)ωA - (μ'y - 1)ω'A'

Dispersion without Average Deviation

For a combination to produce dispersion without average deviation, the net average deviation should be zero.

(μy - 1)A = (μ'y - 1)A'

In this case, the net angular dispersion is:

δ = (μy - 1)A (ω - ω')

Average Deviation without Dispersion

For a combination to produce average deviation without dispersion, the net dispersion should be zero.

(μy - 1)ωA = (μ'y - 1)ω'A'

or

(μv - μr)A = (μ'v - μ'r)A'

The net average deviation is:

d = (μy - 1)A [1 - ((μ'y - 1)A') / ((μy - 1)A)]

Spectrum

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A spectrum is the result of dispersing light from a source, separating it into its constituent wavelengths. A pure spectrum is one where each wavelength occupies a distinct spatial position, achieved with parallel incident light and focused dispersed light.
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Spectrum

A spectrum is a collection of dispersed light, showing its wavelength composition. It is formed when light from a source is dispersed by a prism or other dispersing element, separating light of different wavelengths.

Pure and Impure Spectrum

In a pure spectrum, each wavelength occupies a specific spatial position without overlapping. This provides a sharp impression of each color.

To obtain a pure spectrum, the following conditions must be met:

  1. The light beam incident on the dispersing element should be parallel.
  2. The dispersed light should be focused so that all rays of a particular wavelength are collected at one place.

These conditions can be approximated using a setup with a narrow slit, achromatic lenses, and a dispersing element.

Dispersion of Light

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Dispersion is the phenomenon of splitting of white light into its constituent colors. Dispersive power quantifies the ability of a material to separate colors.
<ul><li>Dispersive power (ω) = (n<sub>v</sub> - n<sub>r</sub>) / (n<sub>y</sub> - 1) where n<sub>v</sub>, n<sub>r</sub>, and n<sub>y</sub> are the refractive indices for violet, red, and yellow light, respectively.</li></ul>
  • Dispersive Power: A measure of the angular dispersion produced by a prism relative to the average deviation. It depends on the refractive indices of the prism material for different colors.
  • Combination of Prisms: Prisms made of different materials can be combined to achieve dispersion without deviation or deviation without dispersion. This is used in achromatic lenses.

Spectra

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Line spectra provide information about atoms, while band spectra provide information about molecules.
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  • Line Spectra: These spectra are characteristic of individual atoms. The wavelengths of the lines are related to the energy levels within the atom.
  • Band Spectra: These spectra are characteristic of molecules and arise from the vibrational and rotational energy levels of the molecule.

Lenses

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Focal length of converging lens depends on the wavelength. Smaller the wavelength, smaller the focal length.
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  • Focal Length and Wavelength: The focal length of a lens depends on the refractive index of the lens material, which in turn depends on the wavelength of light. Since refractive index is greater for violet light than for red light, the focal length for violet light is less than that for red light (fv < fr).

Achromatic Lens

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An achromatic lens is designed to minimize chromatic aberration by combining lenses of different materials to correct for dispersion.
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  • Achromatic Lens: Incident light is passed through a narrow slit placed in the focal plane of an achromatic lens to produce a pure spectrum. The narrow slit produces less diffraction and ensures a more parallel beam of light.