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Ray optics formulas

Refraction

The phenomenon of change in the path of light when it passes from one medium to another

Laws of refraction

First

The incident ray, the refracted ray and the normal at the point of incidence all lie in the same plane

Second

Sin i / Sin r = μ [refractive index of the second medium with respect to the first medium]

Also μ = c/v or v₁/v₂

μ = Real depth / Apparent depth

Light enters medium a, crosses medium b and then leaves from medium c, then ^aμ_c = ^aμ_b x ^bμ_c

Total internal reflection

When light travelling from denser to rarer medium is incident at an angle greater than the critical angle, it is reflected back in the denser medium.

Conditions for total internal reflection

Light should travel from denser to rarer medium.

Angle i > angle i_c where i_c is the critical angle.

Critical angle

When light travels from denser to rarer medium, then that angle of incidence for which angle of refraction is 90^o

Refraction at spherical refracting surface

Rarer to denser medium

-μ₁/u + μ₂/v = (μ₂ - μ₁)/R

where μ₁ and μ₂ are refractive indices of rarer and denser mediums respectively

R is the radius of curvature of the spherical surface.

Denser to rarer medium

-μ₂/u + μ₁/v = (μ₁ - μ₂)/R

Lens maker's formula

1/f = (μ - 1)(1/R₁ - 1/R₂)

Where μ is the refractive index of material of the lens

R₁ and R₂ are the radii of curvature of the two surfaces of the lens.

Lens formula

1/v - 1/u = 1/f

Linear magnification

m = h_i/h_o = v/u

Power of a lens

P = 1/f if f is in meters Units of P: Dioptre D

Combination of two thin lenses

Lenses in contact

1/F = 1/f₁ + 1/f₂ => P = P₁ + P₂ and m = m₁ x m₂

Lenses separated by a finite distance

1/F = 1/f₁ + 1/f₂ - d/f₁f₂

Refraction through a prism

Angle of deviation

δ = (μ - 1) A for A < 10^o (thin prism)

δ = (i₁ + i₂) - A for A > 10^o

here i₁ and i₂ are angles of incidence and emergence

Angular dispersion

δ_v - δ_r = (μ_v - μ_r)A

A + δ = i + e

Prism formula

μ = (Sin(A + δ_m/2)) / Sin A/2

where δ_m is angle of minimum deviation

Dispersive power

w = Angular dispersion/Mean dispersion = (μ_v - μ_r)/(μ - 1)

v and r refer to violet and red colors

μ refers to mean color wavelength (yellow)

Magnifying power

Simple microscope

m = 1 + D/f where D = least distance of distinct vision = 2.5 cm

Compound microscope

The ratio of angle subtended at the eye by the final image to the angle subtended at the eye by the object where both the final image and object are situated at the least distance of distinct vision.

m = L/f_o[1 + (D/f_e)]

L is the length of the microscope tube

f_o is the focal length of objective

f_e is the focal length of the eye piece

Astronomical telescope

Normal adjustment

The final image is formed at infinity

Magnifying power of an astronomical telescope in normal adjustment is defined as the ratio of the angle subtended at the eye by the final image to the angle subtended at the eye by the object directly when the final image and the object both lie at infinite distance from the eye.

m = f_o/-f_e

Final image at least distance of distinct vision

It is defined as the ratio of the angle subtended at the eye by the final image at the least distance of distinct vision to the angle subtended at the eye by the object at infinity, when seen directly

m = (f_o/f_e)(1 + f_e/D)

Cassegrainian reflecting type telescope

m = f_o/f_e = (R/2)/f_e

Resolving power of a microscope

Resolving power = 1/d = (2μSinθ)/λ

where μ is the refractive index of the medium

λ is the wavelength of light

θ is half angle of the cone of light from thee point object to the objective lens

Resolving power of a telescope

Resolving power = 1/dθ = D/1.22λ

where D is the diameter of the object lens

λ is the wave length of light

Laws of reflection

Angle i = angle r

Incident ray, reflected ray and the normal at the point of incidence all lie in the same plane

Mirror formula

1/v + 1/u = 1/f

m = h_i/h_o= -v/u