MCAT Physics and Math - Light and Optics

470

electromagnetic spectrum

describes the full range of frequencies and wavelengths of electromagnetic waves

wavelengths

units: mm (10^–3 m), μm (10^–6 m), nm (10^–9 m), and Å (ångström, 10^–10 m)

radio waves

long wavelength, low frequency, low energy

10⁹–1 m

microwaves

1 m–1 mm

infrared

1 mm–700 nm

visible light

only part of the spectrum that is perceived as light by the human eye

700–400 nm (→ ROY G BIV)

ultraviolet

400–50 nm

X-rays

50–10^–2 nm

gamma rays

short wavelength, high frequency, high energy

<10^–2 nm

Electromagnetic waves

transverse waves because the oscillating electric and magnetic field vectors are perpendicular to the direction of propagation and to each other

speed of light (c)

all electromagnetic waves travel at the same speed

3.00 × 10⁸ m/s

c = fλ

where c is the speed of light in a vacuum and, to a first approximation, also in air, f is the frequency, and λ is the wavelength

blackbody

ideal absorber of all wavelengths of light, which would appear completely black if it were at a lower temperature than its surroundings

color of an object

dependent on the color of light that it reflects/does not absorb

opponent process theory

white = reflects all

balck = absorbs all

rectilinear propagation

light travels through a homogeneous medium in a straight line

Geometrical optics

explains reflection and refraction, as well as the applications of mirrors and lenses

Reflection

rebounding of incident light waves at the boundary of a medium; not absorbed into the second medium; rather, they bounce off of the boundary and travel back through the first medium

law of reflection

θ₁ = θ₂

where θ₁ is the incident angle and θ₂ is the reflected angle, both measured from the normal

normal

line drawn perpendicular to the boundary of a medium

real mirror images

the light actually converges at the position of the image; the ability of the image to be projected onto a screen

virtual mirror images

the light only appears to be coming from the position of the image but does not actually converge there

plane mirrors

flat reflective surfaces; cause neither convergence nor divergence of reflected light rays; always create virtual images; same distance ‘behind’ as in front; conceptualized as spherical mirrors with an infinite radius of curvature

i = –o

r = f = ∞

Spherical mirrors

mirror can be considered a spherical cap or dome taken from a much larger spherically shaped mirror; associated center of curvature (C) and a radius of curvature (r)

center of curvature (C)

point on the optical axis located at a distance equal to the radius of curvature from the vertex of the mirror; center of the spherically shaped mirror if it were a complete sphere

radius of curvature (r)

distance of actual mirrored surface from center of curvature

concave

inside of a sphere to its surface; ‘inward’ curve

convex

outside the sphere; ‘outward’ curve

concave/converging mirror

center of curvature and the radius of curvature are located in front of the mirror; cause parallel incident light rays to converge

convex/diverging mirror

center of curvature and the radius of curvature are behind the mirror; cause parallel incident light rays to diverge

focal length (f)

distance between the focal point (F) and the mirror

f = r/2

where the radius of curvature (r) is the distance between C and the mirror

Key Variables in Miror Optics

1/f = 1/o + 1/i = 2/r

focal length

object distance

image distance

radius of curvature

image distance

image has a positive distance (i > 0), real image - in front of mirror

image has a negative distance (i < 0), virtual image - behin dmirror

magnification (m)

dimensionless value that is the ratio of the image distance to the object distance; ratio of the size of the image to the size of the object

negative signifies an inverted image, while a positive value signifies an upright image

|m| < 1, the image is smaller than the object (reduced); |m| > 1, the image is larger than the object (enlarged)

m = i/o

ray diagram

useful for getting an approximation of where an image is; real vs. virtual, inverted vs. upright, and magnified vs. reduced

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 same angle relative to normal

axis

the normal passing through the center of the mirror

focal point (F)

object at this point no image is formed because the reflected light rays are parallel to each other

Refraction

bending of light as it passes from one medium to another and changes speed

Snell’s Law

n1 sin θ1 = n2 sin θ2

where n1 and θ1 refer to the medium from which the light is coming and n2 and θ2 refer to the medium into which the light is entering

when light enters a medium with a higher index of refraction (n2 > n1), it bends toward the normal (sin θ2 < sin θ1; therefore, θ2 < θ1)

when light travels into a medium where the index of refraction is smaller (n2 < n1), the light will bend away from the normal (sin θ2 > sin θ1; therefore, θ2 > θ1).

speed of light in medium

n = c/v

where c is the speed of light in a vacuum, ν is the speed of light in the medium, and n is the index of refraction of the medium

index of refraction (n)

dimensionless quantity affects speed of light through specific medium

critical angle (θc)

As the incident angle is increased, special incident angle is reached, for which the refracted angle θ2 equals 90 degrees

Total internal reflection

phenomenon in which all the light incident on a boundary is reflected back into the original material, results with any angle of incidence greater than the critical angle, θc

lenses

refract light with two surfaces; two focal points, one on each side

converging lens

thicker at the center

farsighted

diverging lens

thinner at the center

nearsighted

Key Variables in Lens Optics

1/f = 1/o + 1/i

m= -i/o

lensmaker’s equation

non negligible thick lenses

where n is the index of refraction of the lens material, r1 is the radius of curvature of the first lens surface, and r2 is the radius of curvature of the second lens surface

lens ray diagram

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

power (P)

P= 1/f

where f (the focal length) is in meters

measured in diopters

hyperopia

farsightedness

myopia

nearsightedness

Bifocal lenses

corrective lenses that have two distinct regions; one that causes convergence of light to correct for farsightedness (hyperopia) and a second that causes divergence of light to correct for nearsightedness (myopia)

Lenses in contact

series of lenses with negligible distances between them.

Power is additive

magnification is mulltiplicative

aberrations

visual errors

Spherical aberration

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

dispersion

various wavelengths of light separate from each other

ex. prism

Chromatic aberration

dispersive effect within a spherical lens; rainbow halo around images

Diffraction

spreading out of light as it passes through a narrow opening or around an obstacle

Interference

between diffracted light rays lead to characteristic fringes in slit–lens and double-slit systems; displacements of the waves add together

Single Slit difraction

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)

Slit–Lens System

bright central fringe with alternating dark and bright fringes on each side; central bright fringe (maximum) is twice as wide as the bright fringes on the sides; as the slit becomes narrower, the central maximum becomes wider; dark fringes (minima) is given by a sin θ = nλ

Multiple Slits

famous double-slit experiment, Thomas Young showed that the diffracted rays of light emerging from two parallel slits can interfere with one another; light as a wave

constructive interference between the two light waves appear as bright fringes (maxima); light waves interfere destructively, dark fringes (minima) (d sin θ = (n + 1/2)λ)

where d is the distance between the two slits, θ is the angle between the line drawn from the midpoint between the two slits to the dark fringe and the normal, n is an integer indicating the number of the fringe, and λ is the wavelength of the incident wave

Diffraction gratings

multiple slits arranged in patterns; can create colorful patterns similar to a prism

ex. grooves on a CD or DVD

thin film diffraction

light waves reflecting off the external surface of the film interfere with light waves reflecting off the internal surface of the film; interference here is not between diffracted rays, but between reflected rays

ex. soap bubbles or oil puddles in wet parking lots

X-Ray Diffraction

bending of light rays to create a model of molecules; complex two-dimensional image.

Plane-polarized (or linearly polarized) light

light in which the electric fields of all the waves are oriented in the same direction, electric field vectors are parallel

optical activity

causes plane-polarized light to rotate clockwise or counterclockwise by a given number of degrees relative to its concentration due to the presence of chiral centers

specific rotation

rotate clockwise or counterclockwise by a given number of degrees relative to its concentration

polarizers

allow only light with an electric field pointing in a particular direction to pass through

second polarizer is turned so that its axis is perpendicular, no light gets through at all

Circular Polarization

rarely seen natural phenomenon that results from the interaction of light with certain pigments or highly specialized filters; uniform amplitude but a continuously changing direction