MCAT Physics & Math - Chapter 8 - Light & Optics

Blackbody

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

Law of Reflection

- Both angles are measured from the normal - a line drawn perpendicular to the boundary of a medium.

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. This creates an area of multiple images with very slightly different image distances at the edge of the image, which appears blurry.

Magnification

- A dimensionless value that is the ratio of the image distance to the object distance. - By extension, the magnification also gives the ratio of the size of the image to the size of the object.

Chromatic Aberration

- A dispersive effect within a spherical lens. - Depending on the thickness and curvature of the lens, there may be significant splitting of white light, which results in a rainbow halo around images. - This is corrected for in visual lenses with special coatings that have different dispersive qualities from the lens itself.

Ray Diagrams - Convex Mirrors

- A single diverging mirror forms only a virtual, upright and reduced image, regardless of the position of the object. - The further away the object, the smaller the image will be.

Total Internal Reflection

- All the light incident on a boundary is reflected back into the original material. - Results when any angle of incidence is greater than the critical angle.

Spherical Mirrors - Center of Curvature - Radius of Curvature

- Come in two varieties: concave and convex. - Mirror can be considered a spherical cap or dome taken from a much larger spherically-shaped mirror. - Have an associated center of curvature (C) and a radius of curvature (r). - The center of curvature is a point on the optical axis located at a distance equal to the radius of curvature from the vertex of the mirror - the center of curvature would be the center of the spherically-shaped mirror if it were a complete sphere.

Concave vs. Convex

- Concave - a surface that has a similar curvature to the interior of a sphere. The center of curvature and the radius of curvature are located in front of the mirror. - Convex - a surface that has a similar curvature to the exterior of a sphere. The center of curvature and the radius of curvature are behind the mirror.

Converging vs. Diverging Mirrors

- Concave mirrors are called converging mirrors and convex mirrors are called diverging mirrors because they cause parallel incident light rays to converge and diverge after they reflect, respectively.

Diffraction Gratings

- Consist of multiple slits arranged in patterns.

Speed of Light

- Constant represented by c. - Approximately 3x10^8 m/s. - To a first approximation, electromagnetic waves also travel in air with this speed.

Polarizers

- Filters which allow only light with an electric field pointing in a particular direction to pass through. - If one passes a beam of light through a polarizer, it will only let through that portion of the light parallel to the axis of the polarizer. - If a second polarizer is then held up to the first, the angle between the polarizers' axes will determine how much light passes through. - When the polarizers are aligned, all the light that passes through the first polarizer also passes through the second. - When the second polarizer is turned so that its axis is perpendicular, no light gets through at all.

Multiple Lens System - Not In Contact

- For lenses not in contact, the image of one lens becomes the object of another lens. The image from the last lens is considered the image of the system.

Real Lenses - Lensmaker's Equation

- For lenses where the thickness cannot be neglected, the focal length is related to the curvature of the lens surface and the index of refraction of the lens by the lensmaker's equation: - n is the index of refraction of the lens material. - r1 is the radius of curvature of the first lens surface. - r2 is the radius of curvature of the second lens surface.

Slit-Lens System

- If a lens is placed between a narrow slit and a screen, a pattern is observed consisting of a bright central fringe with alternating dark and bright fringes on each side. - The central bright fringe (maximum) is twice as wide as the bright fringes on the sides, and as the slit becomes narrower, the central maximum becomes wider. - The location of the dark fringes (minima) is given by the formula: - a is the width of the slit. - θ is the angle between the line drawn from the center of the lens to the dark fringe and the axis of the lens. - n is an integer indicating the number of the fringe. - λ is the wavelength of the incident wave. - Bright fringes are halfway between dark fringes.

Dispersion - Prism

- If a source of white light is incident on one of the faces of a prism, the light emerging from the prism is spread out into a fan-shaped beam. - This occurs because violet light has a smaller wavelength than red light and so is bent to a greater extent. - Because red experiences the least amount of refraction, it is always on top of the spectrum; violet, having experienced the greatest amount of refraction is always on the bottom of the spectrum.

Image Distance

- If the image has a positive distance (i>0), it is a real image, which implies that the image is in front of the mirror. - If the image has a negative distance (i<0), it is virtual and thus located behind the mirror.

Real vs. Virtual Images

- In general, images created by a mirror can be either real or virtual. - An image is said to be real if the light actually converges at the position of the image. Real images can be projected onto a screen. - An image is virtual if the light only appears to be coming from the position of the image but does not actually converge there.

Electromagnetic Spectrum

- Includes radiowaves on one end (long wavelength, low frequency, low energy) and gamma rays on the other (short wavelength, high frequency, high energy). - Between the two extremes, in order from lowest energy to highest energy are microwaves, infrared, visible light, ultraviolet and x-rays. - Describes the full range of frequencies and wavelengths of electromagnetic waves.

Multiple Lens System - In Contact

- Lenses in contact are a series of lenses with negligible distances between them. - These systems behave as a single lens with equivalent focal length given by:

Plane-Polarized Light

- Light in which the electric fields of all the waves are oriented in the same direction (their electric field vectors are parallel). - It follows that their magnetic fields vectors are also parallel, but convention dictates that the plane of the electric field identifies the plane of polarization. - Unpolarized light has a random orientation of its electric field vectors.

Power

- Measured in diopters, where f (focal length) is in meters. - P has the same sign as f and is therefore positive for a converging lens and negative for a diverging lens. - People who are nearsighted need diverging lenses, while people who are farsighted need converging lenses. - Bifocal lenses are 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) in the same lens.

Sign Conventions for Mirrors

- On the MCAT, for almost all problems involving mirrors, the object will be placed in front of the mirror. - Thus, the object distance (o) is almost always positive.

Thin Spherical LEnses

- On the MCAT, lenses generally have negligible thickness. - Because light can travel from either side of a lens, a lens has two focal points, with one on each side. The focal length can be measured in either direction from the center. - For thin spherical lenses, the focal lengths are equal, so we speak of just one focal length for the lens as a whole.

Plane Mirrors

- Parallel incident light rays remain parallel after reflection from a plane mirror. Plane mirrors - being flat reflective surfaces - cause neither convergence nor divergence of reflected light rays. - Because the light does not converge at all, plane mirrors always create virtual images. In a plane mirror, the image appears to be the same distance behind the mirror as the object is in front of it. - In other words, plane mirrors create the appearance of light rays originating behind the mirrored surface. Because the reflected light remains in front of the mirror but the image appears behind the mirror, the image is virtual. - Can be conceptualized as spherical mirrors with an infinite radius of curvature.

Plane Mirrors - Image Distance

- Plane mirrors can be thought of as spherical mirrors with infinitely large focal distances. - As such, for a plane mirror, r=f=∞, and the equation becomes 1/o + 1/i = 0 or i=-o. - This can be interpreted as saying the virtual image is at a distance behind the mirror equal to the distance the object is in front of the mirror.

Circular Polarization

- Rarely seen. - Results from the interaction of light with certain pigments or highly specialized filters. - Circularly polarized light has a uniform amplitude but a continuously changing direction, which causes a helical orientation in the propagating wave. - The helix has average electrical field vectors and magnetic field vectors that lie perpendicular to one another, like other waves, with maxima that fall on the outer border of the helix.

Snell's Law - Equation

- Refracted rays of light obey Snell's Law as they pass from one medium to another. - 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. θ is measured with respect to the normal.

Refraction

- The bending of light as it passes from one medium to another and changes speed. - The speed of light through any medium is always less than its speed through a vacuum. - For a given medium the equation is given where c is the speed of light in a vacuum, v is the speed of light in the medium and n is a dimensionless quantity called the index of refraction of the medium. - The index of refraction of a vacuum is 1, by definition; for all other materials, the index of refraction will be greater than 1. - For air, n is essentially equal to 1 because the speed of light in air is extremely close to c.

Real Lenses - Eye

- 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. - At this point, the image has been focused and minimized significantly, but is still relatively blurry. The nervous system processes the remaining errors to provide a crisp view of the world.

Real Vs. Virtual - Mirrors vs. Lenses

- The designations of real and virtual are on opposite sides when comparing mirrors and lenses. - The real side if where light actually goes after interacting with the lens or mirror. - For mirrors, light is reflected and therefore stays in front of the mirror. Hence, for a mirror, the real side is in front of the mirror, and the virtual side is behind the mirror. - For lenses the convention is different; because light travels through the lens and comes out on the other side, the real side is on the opposite side of the lens from the original light source and the virtual side is on the same side of the lens as the original light source. Although the object of a single lens is on the virtual side, this does not make the object virtual. - Objects are real, with a positive object distance, unless they are placed in certain multiple lens systems in which the image of one lens becomes the object for another.

Key Variables in Geometrical Optics

- The focal length (f) is the distance between the focal point (F) and the mirror. - For all spherical mirrors, f=r/2 where the radius of curvature (r) is the distance between C and the mirror. - The distance between the object and the mirror is (o). - The distance between the image and the mirror is (i).

Visible Region

- The only part of the spectrum that is perceived as light by the human eye. - Within this region, different wavelengths are perceived as different colors with violet at one end of the visible spectrum (400 nm) and red at the other (700 nm). - Light that contains all the colors in equal intensity is perceived as white. - The color of an object that does not emit its own light is dependent on the color of light that it reflects. Thus, an object that appears red is one that absorbs all colors of light except red.

Plane-Polarized Light - Stereoisomers

- The optical activity of a compound, due to the presence of chiral centers causes plane-polarized light to rotate clockwise or counterclockwise by a given number of degrees relative to its concentration (specific rotation). - Enantiomers, as nonsuperimposable mirror images, will have opposite specific rotations.

Young's Double-Slit Experiment - Equation

- The positions of dark fringes (minima) on the screen can be found from the equation: - 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. - λ is the wavelength of the incident wave.

Reflection

- The rebounding of incident light waves at the boundary of a medium. - Light waves that are reflected are not absorbed into the second medium; rather, they bounce off of the boundary and travel back through the first medium.

Dispersion

- The speed of light in a vacuum is the same for all wavelengths. However, when light travels through a medium, different wavelengths travel at different speeds. This fact implies that the index of refraction of a medium affects the wavelength of light passing through the medium because the index of refraction is related to the speed of the wave by n=c/v. - It also implies that the index of refraction itself actually varies with wavelength. - When various wavelengths of light separate from each other, this is called dispersion. The most common example of dispersion is the splitting of white light into its component colors using a prism.

Diffraction

- The 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.

Ray Diagrams - Concave Mirrors

- There are three important rays to draw: - For a concave mirror, a ray that strikes the mirror parallel to the axis (the normal passing through the center of the mirror) is reflected back through the focal point (green lines). - A ray that passes through the focal point before reaching the mirror is reflected back parallel to the axis (red lines). - A ray that strikes the mirror at the point of intersection with the axis is reflected back with the same angle measured from the normal (blue lines).

Thin Film Interference

- Thin films may also cause interference patterns because 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.

Electromagnetic Waves

- Transverse waves because the oscillating electric and magnetic field vectors are perpendicular to the direction of propagation. - The electric field and magnetic field are also perpendicular to each other. - Vary in frequency and wavelength, but in a vacuum, all electromagnetic waves travel at the speed of light.

X-Ray Diffraction

- Uses the bending of light rays to create a model of molecules. - Often combined with protein crystallography during protein analysis. - Dark and light fringes do not take on a linear appearance, but rather 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 (sin θ2< sin θ1); therefore, (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 (sinθ2>sinθ1; therefore, 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.

Critical Angle

- When light travels from a medium with a higher index of refraction to a medium with a lower index of refraction, the refracted angle is larger than the incident angle (θ2>θ1); that is, the refracted light ray bends away from the normal. - As the incident angle is increased, the refracted angle also increases, and eventually a special incident angle called the critical angle (θc) is reached, for which the refracted angle θ2 equals 90°. - At the critical angle, the refracted light ray passes along the interface between the two media. - The critical angle can be derived from Snell's law if θ2=90°, such that:

Geometrical Optics - Rectilinear Propagation

- When light travels through a homogenous medium, it travels in a straight line - rectilinear medium. - The behavior of light at the boundary of a medium or interface between two media is described by the theory of geometrical optics.

Young's Double-Slit Experiment

- When waves interact with each other, the displacements of the waves add together - interference. - In his double-slit experiment, Thomas Young showed that the diffracted rays of light emerging from two parallel slits can interfere with one another. - Finding contributed to understanding light as a wave. - When monochromatic light (light of only 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). - In regions where the light waves interfere destructively, dark fringes (minima) appear.

Differences between Lenses & Mirrors:

1. Lenses refract light while mirrors reflect it. 2. When working with lenses, there are two surfaces that affect the light path.

Ray Diagrams for Single Lenses

a) A converging lens is always thicker at the center. b) A diverging lens is always thinner at the center. - The basic formulas for finding image distance and magnification for spherical mirrors also apply to lenses.

Ray Diagrams - Concave Mirrors - Results

a) The object is placed beyond F and the image produced is real, inverted and magnified. b) The object is placed at F and no image is formed because the reflected light rays are parallel to each other. i = ∞. c) The object is placed between F and the mirror, and the image produced is virtual, upright and magnified.