Ray Optics and Optical Instruments

Refraction through a Prism, Dispersion by a Prism

  • Prism is a transparent piece having two refracting surfaces with non zero angle between them.
  • In prism Angle of prism A = r1 + r2

  • In prism Angle of Deviation d = i1 + i2 – A.
  • Condition for minimum deviation in prism i1 = i2 = i3, r1 = r2 = r.
  • Refractive index of prism with respect to surroundings \tt \mu = \frac{\sin \frac{A + D_{m}}{2}}{\sin \frac{A}{2}}
  • Condition for normal incidence and grazing emergence i1 = r1 = 0  i2 = 90°  &  r2 = c = A.
  • For grazing emergence \tt A = C = \sin^{-1} \left(\frac{1}{\mu}\right)
  • Condition for grazing incident and grazing Emergence i1 = i2 = 90°    r1 = r2 = C      A = 2C      d = 180° − 2C
  • In order to have an Emergent ray the maximum angle of prism is “2 c” which is called Limiting angle of prism.
  • Condition for total internal reflection at second face of prism sin (i) = \tt \sqrt{\mu^{2} - 1} (sin A – cos A)
  • Deviation angle for small angled prisms d = (μ t)A
  • Deviation angle for small angled prisms in medium \tt d = \left(\frac{\mu g}{\mu m}t\right) A
  • The separation of composite beam of light into constituent colours is called Dispersion.
  • In Dispersion the angle of refraction is most for red and least for violet (r v < r R)
  • In Dispersion Angle of deviation is most for Violet and least for red (dv > dR)
  • In Dispersion the refractive index is most for violet and least for red (μv > μR)
  • The difference between angles of deviation for any pair of colours is called angular dispersion. θ = δV – δR.
  • Dispersive power of medium (ω) is the ratio between angular dispersion and mean angle of deviation. \tt \omega = \frac{\theta}{\delta y} = \frac{\delta v - \delta R}{\left(\frac{\delta v + \delta R}{2}\right)}
  • The deviation of yellow is taken as Mean deviation of violet and red \tt \omega = \frac{\delta v - \delta R}{\delta y}
  • Angular dispersion o for small angles prisms θ = δv – δR = (μv – μR) A.
  • Dispersive power for small angles prisms \tt \omega = \frac{\theta}{\delta y} = \frac{\delta v - \delta R}{\delta y} = \frac{\mu v - \mu R}{\mu y - 1}
  • Condition for deviation without dispersion. θ + θ1 = 0 (or) ωδ + ω1δ1 = 0.
  • Net deviation = δ + δ1 = (μ − 1)A + (μ1 − 1) A1
  • Condition for Dispersion without deviation is δ + δ1 = 0  (or)   (μ − 1) A + (μ1 − 1) A1 = 0
  • Net dispersion = θ + θ1 = ωδ + ω1δ1
  • Law of refraction at spherical surface \tt \frac{\mu_{2}}{v} - \frac{\mu_{1}}{u} = \frac{\mu_{2} - \mu_{1}}{R}
  • Lens makers formula \tt \frac{1}{f} = \left(\frac{\mu_{2}}{\mu_{1}} - 1\right) \left(\frac{1}{R_{1}} - \frac{1}{R_{2}}\right)
  • For diverging miniseus \tt \frac{1}{f} = \left(\frac{\mu_{2}}{\mu_{1}} - 1\right) \left(\frac{1}{R_{1}} - \frac{1}{R_{2}}\right)
  • For converging minisens \tt \frac{1}{f} = \left(\frac{\mu_{2}}{\mu_{1}} - 1\right) \left(\frac{1}{R_{1}} + \frac{1}{R_{2}}\right)
  • For convex lens \tt \frac{1}{f} = \left(\frac{\mu_{2}}{\mu_{1}} - 1\right) \left(\frac{1}{R_{1}} + \frac{1}{R_{2}}\right)
  • For concave lens \tt \frac{1}{f} = \left(\frac{\mu_{2}}{\mu_{1}} - 1\right) \left(\frac{1}{R_{1}} +\frac{1}{R_{2}}\right)
  • The focal power (P) of a lens is numerically equal to the reciprocal of its focal length \tt P = \frac{1}{f} (diopters)
  • If two lenses are separated by a distance “d” \tt \frac{1}{f} = \frac{1}{f_{1}} + \frac{1}{f_{2}} - \frac{d}{f_{1} f_{2}}
  • P = P1 + P2 - dP1 P2
  • When an Equiconvex lens (f) is cut into two plano convex lenses focal length of each becomes “2f”.
  • One of the surface of Convex lens is silvered. \tt \frac{1}{F} = \frac{1}{fl} + \frac{1}{fm} + \frac{1}{fl} = \frac{2}{fl} + \frac{1}{fm}

  • If plane surface of a Planoconvex lens is silvered \tt \frac{1}{F} = \frac{2}{fl} + \frac{1}{fm}

\tt \frac{1}{F} = \frac{2}{fl} + \frac{1}{\infty}

\tt F = \frac{R}{2 (\mu + 1)}

  • If the spherical surface of a planoconvex lens is silvered

\tt \frac{1}{F} = \frac{2}{fl} + \frac{1}{fm}

\tt F = \frac{R}{2 \mu }

  • Rainbows are formed by dispersion of sunlight falling on raindrops.
  • If the molecules of a medium after absorbing in coming radiations emits in all possible directions this process is called scattering.

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For surface AC \mu = \frac{\sin i}{\sin r_{1}}; For surface AB \ = \frac{1}{\mu} = \frac{\sin r_{2}}{\sin e}