Physics-Intro to Optics, Relativity

Snell's Law

n ₁ Sin [θ ₁ ] = n ₂ Sin [θ ₂ ]
c _n = c / n
λ _n = λ / n
I _θ = I ₀ E ² _θ / E ² ₀ 
Where n is the refraction index of the material

Lens Equation:

1 / d _i + 1 / d _o = 1 / f
m = - d _i / d _o 
Where d _i is the distance to the image and d _o is the distance to the object. m represents the magnification of the object in comparison to the objects actual size.

Lens maker Equation:

1 / f = (n - 1) (1 / R ₁ + 1 / R ₂ )
r = f / 2
f is the focal length, and r is the radius of curvature. The focal length is the distance at which the parallel rays of light travelling through the lens will converge to a point. The radius of curvature represents the radius of the sphere that the lens could make if it was continued to a sphere.

Interference and Diffraction-

Double Slit interference:

I _θ = I ₀ [Cos(δ / 2)] ² 
d Sin[θ] = m λ / Maxima
d Sin[θ] = (m +1/2) λ / Minima

Single Slit diffraction:

I _θ = I ₀ [Sin(Β/2) / (Β / 2)] ² 
D Sin[θ] = (m +1/2) λ / Maxima
Note - θ = 0 is the central and highest peak
D Sin[θ] = m λ / Minima

Single and double slit effects:

I _θ = I ₀ [Sin(Β/2) / (Β / 2)] ² [Cos(δ / 2)] ²

Diffraction grating:

Sin (θ) = m λ / d

R = λ / (Δ λ) = N m
R is the resolving power of the lens

Circular hole diffraction

θ = 1.22 λ / D

Polaroids

I = I ₀ (Cos[θ]) ²

Relativity

Minkowski space-This is the setting in which relativistic calculations are done. Points in Minkowski space are represented by a 4-vector which defines position in x, y, z and time, and is represented by the Minkowski metric, which is defined as:

[-1 0 0 0]
| 0 1 0 0|
| 0 0 1 0|
[ 0 0 0 1]

Invariance of the interval:

(Δs) ² = (Δx) ² - (c Δt) ² 
(Δs') ² = (Δs) ² 
The invariance of the interval is also known as the line element of the minkowski metric.

Lorentz Transformations-

ct = γ ( v x' / c + c t ' )
x = γ ( x' + v t ' )
y = y'
z = z'

This is assuming your working in a frame where there is movement in only one direction, however due to the isotropy of space, you can almost always make this assumption.

γ = 1 / [1 - (v / c) ² ] ^1/2 
This is known as the lorentz factor, which is used in the lorentz transformation.

Velocity addition formulas-
u _parallel = (u _parallel ' + v) / (1 + (u _x ' v / c ² ))
u _{perpendicular} = (u _{perpendicular} ') / [γ (1 + (u _{perpendicular} ' v / c ² ))]

Contraction

L' = L / γ
Where L' is the length of the moving object as observed by the stationary frame.
Δt' = γ Δt
Where t ' is the length of time experienced by an observer in the stationary frame.

Stationary frames are described by no acceleration.

Non constant velocity

τ = ∫ [1 - ( v ² (t) / c ² )] ^1/2 dt

Momentum and Energy

p = γ m v
E = γ m c ² = [m ² c ⁴ + |p| ² c ² ]
E ² = m ² c ⁴ + p ² c ² 
KE = E - m ² c ² = (γ - 1) m c ²