List of formulas related to gemetric optics

Laws Of Reflection

SI: incident ray, IR: reflected ray, i: angle of incidence, i’: angle of reflection

$$ i = i'$$

Refractive Index

n: refractive index of the medium, v: speed of light in a medium, c: speed of light in vacuum (m/s)

$$ n = \frac{c}{v} $$

Law Of Refraction

n₁: refractive index of the medium 1, n₂: refractive index of medium 2, i: angle of incidence, r: angle of refraction, n₂₁: absolute index of refraction

$$ \frac{sin i}{sin r} = n_{21} = \frac{n_2}{n_1} $$ $$ \frac{n_1}{n_2} = \frac{v_1}{v_2} $$ $$ v_1 = c =3.10^8 m/s $$ $$ n_1 = 1 $$ $$ => n_2 = \frac{c}{v_2} $$

Total Reflection Phenomenon

There is no refracted ray if

$$ n_1 > n_2 $$ $$ i > i_c, sin i_c = \frac{n_2}{n_1} $$

Prism

A: apex angle, D: deviation angle, i₁: angle of incidence, i₂: angle of emergent ray

$$ sin i _1 = nsin r _1 $$ $$ sin i_2 = nsin r_2 $$ $$ A = r_1 + r_2 $$ $$ D = i_1 + i_2 - A $$ $$ \text{Condition to have refractive ray} $$ $$ A \leq 2i_{gh} $$ $$ i \geq i_o $$ $$ sin i_o = nsin(A - \tau) $$ $$ \text{When angle of incidence is minimum} $$ $$ d - > D_{min} : i_1 = i_2 = i $$ $$ r_1 = r_2 = \frac{A}{2} $$ $$ sin \frac{D_{min} + A}{2} = nsin \frac{A}{2} $$ $$ => D_{min} = 2i - A $$

Thin Lens

f: focal length (converging len f>0; diverging len f<0), d: object distance from lens center, d’: image distance from lens center

$$ \frac{1}{f} = \frac{1}{d} + \frac{1}{d'} => f = \frac{d.d'}{d + d'} $$ $$ d = \frac{d'.f}{d' - f'} ; {d'} = \frac{df}{d - f} $$

Magnification Factor

h_I ; image height, h_o: object height

$$ k = \frac{h_1}{h_o} = - \frac{d'}{d} = \frac{-f}{d - f} = \frac{d' - f}{f} $$

Power Of The Lens

n: refractive index of the lens material, R₁: radius of curvature of the lens surface closest to the light source, R₂: radius of curvature of the lens surface farthest from the light source

$$ D = \frac{1}{f} = (n - 1) (\frac{1}{R_1} + \frac{1}{R_2}) $$