Physics and Math 8
The eye
A complex refractive instrument that uses real lenses The cornea acts as the primary source of refractive power because the change in refractive index from air is so significant Then light is passed through an adaptive lens that can change its focal length before reaching the vitreous humor It is further diffused through layers of retinal tissue to reach the rods and cones The image has been focused but is still blurry until the nervous system processes the remaining error to provide a clear image
Magnification (m)
A dimensionless value that is the ratio of the image distance to the object distance negative magnification signifies an inverted image positive value signifies an upright image If |m|<1, then the image is smaller than the object (reduced) If |m|>1, the image is larger than the object (magnified) If |m|=1, then the image is the same size as the object.
Chromatic Aberration
A dispersive affect within a spherical lends that leads to a rainbow halo at the edge of the image Ex. It is seen on car windows and glasses (light is split to colors).
Total Internal Reflection
A phenomenon in which all the light incident on a boundary is reflected back into the original material, resulting with any angle of incidence greater than the critical angle (θc) Occurs as the light moves from a medium with a higher refractive index to a medium with a lower one
Real and Virtual Images
An image is said to be real if the light actually converges at the position of the image An image is said to be virtual is the light only appears to be coming from the position of the image, but doesn't actually converge there A real image can be projected onto a screen
Spherical Mirrors
Concave • inside of sphere looking out • the C and r are in front of the mirror • Converging mirrors Convex • looking at the outside of a sphere • the C and r are behind the mirror • Diverging mirrors Center of curvature (C), a point on the optical axis located at a distance equal to the Radius of curvature (r) from the vertex in the mirror Focal Length (f): the distance between the focal point (F) and the mirror f = r/2 o = distance between the object and the mirror I = distance between the image and the mirror
Diffraction Gratings
Consist of multiple slits arranged in patterns Can create colorful patterns similar to a prism as the different wavelengths interfere in characteristic patterns Ex. a CD or DVD Thin films may also cause interference patterns Ex. soap bubbles or oil puddles
Constructive and Destructive Interference
Constructive interference: Crests align perfectly (in-phase) resulting in a combined wave with a crest that equals the sum of the heights of the original crests - Crest aligns with crest - Valley aligns with valley Destructive interference: Waves are not perfectly aligned, and therefore the crest of one wave will be dragged down by the trough of another wave. - The resulting combined waves will have crests and troughs that are smaller than either of the incoming waves. - If two waves are shifted by exactly half a wavelength (180° out of phase) when they merge they will cancel each entirely (which is the premise for noise-canceling headphones) - Crest aligns with valley - Valley aligns with crest
To find where an image is (mirrors)
Draw the following rays and find a point where any two intersect (the tip of an image) If they do not converge, extend them to the other side of the mirror, creating a virtual image • Ray parallel to axis → reflects back through focal point • Ray through focal point → reflects back parallel to axis • Ray to center of mirror → reflects back at the same angle relative to normal
To find where an image is (lens)
Draw the following rays and find a point where any two intersect (the tip of an image) If they do not converge, extend them to the same side of the lens from which the light came, creating a virtual image • Ray parallel to axis → refracts through focal point of front face of the lens • Ray through or toward focal point before reaching lens → refracts parallel to axis • Ray to center of lens → continues straight through with no refraction
Huygen's Principle
Every point on any wave front can be regarded as a new point source of secondary waves mλ = w⋅sinθ m = order of the DESTRUCTIVE POINTS which are denoted by m=1,2,3,4 counting up from the center line, and m≠0 λ = wavelength w = width of the slit θ = angle of the triangle made from the center line and the dark spot on the wall
Geometrical Optics
Explains reflection and refraction
Plane Mirrors
Flat reflective surfaces which cause neither convergence nor divergence of reflected light rays Hence, images produced are upright, virtual, reversed and seem to be as far behind the mirror as the object is in front Ex. Mirrors
Slit-Lens System
If a lens is placed between a narrow slit and screen, a pattern of bright and dark fringes are seen surrounding a large central light fringe from diffraction As the slit gets smaller the central max gets larger The central bright fringe (maximum) is 2x as wide as the bright fringes on the sides and as the slit gets smaller the central max gets wider equation: asinθ=nλ • Gives the location of the dark fringes (minima) a = width of the slit, θ= angle between the line of the center lens to the dark fringe and the axis of the lens, n = number of the fringe, λ = incident wave
Multiple Slits
Interference: when waves interact with each other, the displacements of the waves add together When monochromatic light (light of one wavelength) passes through the slits, an interference pattern is observed on a screen placed behind the slits Regions of constructive interference between the two light waves appear as bright fringes (maxima) on the screen and in regions where the light waves interfere destructively, dark fringes (minima) appear Position of dark fringes equation: dsinθ = (n + 1/2)λ d = distance between two slits θ = angle between the line drawn from the midpoint between the two slits to the dark fridge and the normal n = number of fringe λ = wavelength
Multiple Lens Systems
Lenses in contact are a series of lenses with negligible distances between them Lenses not in contact, the image of one lens becomes the object of another lens Magnification is m = m1 x m2 x m3 x ... x mn
Real Lenses
Lenses with non-negligible thickness With real lenses, the focal length is related to the curvature of the lens surfaces and the index of refraction of the lens by the Lensmaker's equation
Plane-Polarized (Linearly polarized) Light
Light in which the electric fields of all the waves are oriented in the same direction (their electrical and magnetic fields are parallel) Unpolarized light has a random orientation of its electric field vectors (sunlight and light bulbs) Polarizers: filters that allow only light with an electric field pointing in a particular direction to pass through If a beam of light passes through a polarizer, it will only let through that portion of the light parallel to the polarizer's axis When 2 polarizers are aligned, all the light that passes through the 1st passes through the 2nd If the 2nd polarizer is turned perpendicular to the 1st, no light can get through
Real vs Virtual
Mirrors: the real side is in front of the mirror and the virtual side is behind the mirror Lenses: the real side is on the opposite side of the lens and the virtual side is on the same side of the lens Converging species: positive focal lengths and radii of curvature Diverging species: negative focal lengths and radii of curvature Lens have 2 focal lengths and 2 radii of curvature because they have 2 surfaces • For a thin lens where thickness is negligible, the sign of the focal length and radius of curvature are given based on the first surface the light passes through
Power
Positive for converging lens and negative for a diverging lens Nearsighted (can see near objects clearly) is corrected by diverging lenses Farsighted (can see distant objects clearly) is corrected by converging lens
Diffuse Reflection
Reflection that occurs when parallel rays of light hit a rough surface and all reflect at different angles
Circular Polarization
Results from the interaction of light with certain pigments or special filters Uniform amplitude but constantly changing direction
Spherical Aberration
Spherical mirrors and lenses are imperfect, so they are subject to errors or aberrations Spherical aberration: a blurring of the periphery of an image as a result of inadequate reflection of parallel beams at the edge of a mirror or inadequate refraction of parallel beams at the edge of a lens Creates an are of multiple images with very slightly different image distances at the edge of the image, which appears blurry
Refraction
The bending of light as it passes from one medium to another and changes speed Speed of light in air ~ c = 3.00 x 10^8 m/s When light is any medium besides a vacuum, its speed is less than c n (index of refraction) = c/v
Thin Spherical Lenses
The focal lengths are equal A converging lens is always thicker at the center and a diverging lens is always thinner at the center 1/f = 1/o + 1/i and m = -i/o
Color and the Visible Spectrum
The part of the spectrum that is perceived as light by the human eye Wavelengths of 400 nm (violet) to 700 nm (red) The color of an object that doesn't emit its own light is dependent on the color of light it reflects (an object that appears red absorbs all colors of light except red) White: light that contains all the colors in equal intensity Blackbody: an ideal absorber of all wavelengths of light, which would appear completely black
Reflection
The rebounding of incident light waves at the boundary of a medium (instead of being absorbed, they bounce off)
Diffraction
The spreading-out effect of light when it passes through a narrow opening or around an obstacle
Electromagnetic Waves
Transverse waves that consist of an oscillating electric field and an oscillating magnetic field (the two fields are perpendicular to one another and the direction of propagation of the wave)
Lenses
Two surfaces that affect the light path Light is refracted twice as it passes from air to lens and from lens back to air Has two focal points, with one on each side (focal length can be measured from either direction)
UV NO IR
Upright images are always Virtual NO image is formed when the object is a focal length away Inverted images are always Real
Young's Double Split Equation
Used to determine Δx (the difference in pathlength between two waves)
X-Ray Diffraction
Uses bending of light rays to create a model of molecules Often combined with protein crystallography during protein analysis Creates a complex 2D image
Snell's Law
When light enters a medium with a higher index of refraction (n2 > n1) it bends toward the normal (θ2 < θ1) If the light travels into a medium where the index of refraction is smaller (n2 < n1), the light will bend away from the normal (θ2 > θ1)
Single Slit
When light passes through a narrow opening (an opening with a size that is on the order of light wavelengths), the light waves seem to spread out (diffract) As the slit is narrowed, the light spreads out more
Rectilinear Propagation
When light travels through a homogenous medium, it travels in a straight line
Dispersion
When various wavelengths of light separate from each other Ex. splitting white light into a spectrum using a prism Occurs because different colors have different wavelengths and so they refract more when they have smaller wavelengths (violet has the most refraction and red has the least) Smaller wavelength light = larger index of refraction = smaller theta2 = bends more
Sign Conventions for Lenses
concave mirrors and convex lenses are both converging and thus have similar properties (same with convex mirrors and concave lenses)
Distance Relationship
r = f = infinity, so the equation becomes 1/o + 1/i = 0 or i = -o
Law of Reflection
the angle of incidence is equal to the angle of reflection Normal: a line drawn perpendicular to the boundary of a medium in which all angles are measured from
Electromagnetic Spectrum
the range of wavelengths or frequencies found in EM waves Radio waves (long wavelength, low frequency, low energy) → microwaves → infrared → visible light → ultraviolet → x-rays → gamma rays (short wavelength, high frequency, high energy)