LIGHT: Geometric Optics Ch 23

Reflection off of smooth surfaces such as mirrors or a calm body of water leads to a type of reflection known as specular reflection.

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Reflection off of rough surfaces such as clothing, paper, and the asphalt roadway leads to a type of reflection known as diffuse reflection.

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There are several interesting applications of this distinction between specular and diffuse reflection. One application pertains to the relative difficulty of night driving on a dry asphalt roadway compared to a wet asphalt roadway.

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The Law of Reflection

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In the diagram, the ray of light approaching the mirror is known as the incident ray

The ray of light which leaves the mirror is known as the reflected ray

At the point of incidence where the ray strikes the mirror, a line can be drawn perpendicular to the surface of the mirror; this line is known as a normal line

This law of reflection can be derived from Fermat's principle.

Fermat's Principle: Light follows the path of least time

The path-length L from A to B is

Since the speed is constant, the minimum time path is simply the minimum distance path. This may be found by setting the derivative of L with respect to x equal to zero.

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This derivation makes use of the calculus of maximum-minimum determination, the derivative of a square root, and the definitions of the triangle trig functions.

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The normal line divides the angle between the incident ray and the reflected ray into two equal angles. The angle

between the incident ray and the normal is known as the angle of incidence.

The angle between the reflected ray and the normal is known as the angle of reflection.

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Object distance = Image distance

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Lets try an experiment!

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Image Formation in Plane Mirrors

An image is a position in space where all the reflected light appears to diverge from. Virtual images are images which are formed in locations where light does not actually reach. Whenever a mirror (whether a plane mirror or otherwise) creates an image which is virtual, it will be located behind the mirror where light does not really pass.

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Besides the fact that plane mirror images are virtual, there are several other characteristics which are worth noting.

1. If you raise your left hand, you will notice that the image raises its right hand. If you raise your right hand, the image raises its left hand. This is termed left-right reversal.

3. The image is said to be upright, as opposed to inverted.

2. For plane mirrors, the object distance (often represented by the symbol do) is equal to the image distance (often represented by the symbol di).

4. the dimensions of the image are the same as the dimensions of the object.

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Images formed by plane mirrors are virtual, upright, left-right reversed, the same distance from the mirror as the object's distance, and the same size as the object.

The ratio of the image dimensions to the object dimensions is termed the magnification.

Plane mirrors produce images which have a magnification of 1.

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CURVED OR SPHERICAL MIRRORS:

There are two types –

a. Concave

b. Convex

Beginning a study of spherical mirrors demands that you first become acquainted with some terminology which will be periodically used.

Principal axis Center of Curvature (C) Vertex (A)

Focal Point (F) Radius of Curvature (R) Focal Length (f)

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The point in the center of sphere from which the mirror was sliced is known as the center of curvature

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The point on the mirror's surface where the principal axis meets the mirror is known as the vertex

The vertex is the geometric center of the mirror.

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Midway between the vertex and the center of curvature is a point known as the focal point

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The distance from the vertex to the center of curvature is known as the radius of curvature. The radius of curvature is the radius of the sphere from which the mirror was cut.

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Finally, the distance from the mirror to the focal point is known as the focal length

Since the focal point is the midpoint of the line segment adjoining the vertex and the center of curvature, the focal length would be one-half the radius of curvature.

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The focal point is the point in space at which light incident towards the mirror and traveling parallel to the principal axis will meet after reflection.

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In our study of concave mirrors, we are going to assume that all light-rays which strike a mirror parallel to its principal axis (e.g., all rays emanating from a distant object) are brought to a focus at the same point. This is only an approximation. It turns out that as rays from a distant object depart further from the principal axis of a concave mirror they are brought to a focus ever closer to the mirror. This lack of perfect focusing of a spherical mirror is called spherical aberration. The approximation in which we neglect spherical aberration is called the paraxial approximation.3 Likewise, the study of image formation under this approximation is known as paraxial optics. This field of optics was first systematically investigated by the famous German mathematician Karl Friedrich Gauss in 1841. (3)

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When the problematic portion of the mirror is disabled so that it can no longer focus (or mis-focus) light, the image appears more focused

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The only type of mirror which does not suffer from spherical aberration is a parabolic mirror

A ray traveling parallel to the principal axis of a parabolic mirror is brought to a focus at the same point, no matter how far the ray is from the axis. Since the path of a light-ray is completely reversible, it follows that a light source placed at the focus of a parabolic mirror yields a perfectly parallel beam of light, after the light has reflected off the surface of the mirror.

A 40' deep focus parabolic mirror generating 88KW worth of steam power (2)

The receiving dishes of radio telescopes are generally parabolic: they reflect the incoming radio waves from (very) distant astronomical sources and bring them to a focus at a single point, where a detector is placed.

A car headlight consists of a light-bulb placed at the focus of a parabolic reflector. The use of a parabolic reflector enables the headlight to cast a very straight beam of light ahead of the car.

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For a concave mirror, the normal at the point of incidence on the mirror surface is a line which extends through the center of curvature. Once the normal is drawn the angle of incidence can be measured and the reflected ray can be drawn with the same angle

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Ray diagrams can be used to determine the image location, size, orientation and type of image formed of objects when placed at a given location in front of a concave mirror.

From the geometry of the spherical mirror, note that the focal length is half the radius of curvature

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The mirror equation expresses the quantitative relationship between the object distance (do), the image distance (di), and the focal length (f).

The Magnification equation relates the ratio of the image distance and object distance to the ratio of the image height (hi) and object height (ho).

These two equations can be combined to yield information about the image distance and image height if the object distance, object height, and focal length are known. (1)

As in the case of lenses, the cartesian sign convention is used here, and that is the origin of the negative sign above. The radius r for a concave mirror is a negative quantity (going left from the surface), and this gives a positive focal length, implying convergence.

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If the object is outside the focal length, a concave mirror will form a real, inverted image.

                                                                      

                                                                           

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If an object is placed inside the focal length of a concave mirror, and enlarged virtual and erect image will be formed behind the mirror.

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Converging raysDivergingrays

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When the object is located beyond the center of curvature (C), the image is located between the center of curvature (C) and the focal point (F). On the other hand, when the object is located between the center of curvature (C) and the focal point (F), the image is located beyond the center of curvature (C).

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 The best means of summarizing this relationship is to divide the possible object locations into five general areas or points:

Case 1: the object is located beyond the center of curvature (C)

Case 2: the object is located at the center of curvature (C)

Case 3: the object is located between the center of curvature (C) and the focal point (F) Case 4: the object is located at the focal point

(F) Case 5: the object is located in front of the focal point (F)

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Case 1: The object is located beyond C

Case 2: The object is located at C

Case 3: The object is located between C and F

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Case 4: The object is located at F

Case 5: The object is located in front of F

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Nine different object locations are drawn and labeled with a number; the corresponding image locations are drawn in blue and labeled with the identical number.

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A convex mirror forms a virtual image.

The virtual image that is formed will appear smaller and closer to the mirror than the object

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The center of that original sphere is known as the center of curvature (C) and the line which passes from the mirror's surface through the sphere's center is known as the principal axis. The mirror has a focal point (F) which is located along the principal axis, midway between the mirror's surface and the center of curvature. Note that the center of curvature and the focal point are located on the side of the mirror opposite the object - behind the mirror. Since the focal point is located behind the convex mirror, such a mirror is said to have a negative focal length value.

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A convex mirror is sometimes referred to as a diverging mirror due to its ability to take light from a point and diverge it.

After reflection, the light rays diverge; subsequently they will never intersect on the object side of the mirror. For this reason, convex mirrors produce virtual images which are located somewhere behind the mirror.

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Any incident ray traveling parallel to the principal axis on the way to a convex mirror will reflect in a manner that its extension will pass through the focal

point.

Any incident ray traveling towards a convex mirror such that its extension passes through the focal point will reflect and travel parallel to the principal axis.

A

B

Ray Diagrams - Convex Mirrors

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Image Characteristics for Convex Mirrors

(a) located behind the convex mirror

(b) a virtual image (c) an upright image

(d) reduced in size

Unlike concave mirrors, convex mirrors always produce images which share these characteristics. The location of the object does not effect the characteristics of the image.

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Another characteristic of the images of objects formed by convex mirrors pertains to how a variation in object distance effects the image distance and size.

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The Mirror Equation - Convex Mirrors

The Magnification equation

Sample Problem (1)

Lenses that cause an incoming plane wave to bend towards the axis through its middle are called converging or convex lenses.

Concave lenses, on the other hand, are thicker at their edges than in the middle; they cause an incoming plane wave to bend away from its central axis and are hence also known as a diverging lenses.

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1. http://www.glenbrook.k12.il.us/gbssci/phys/Class/refln/u13l1c.html

2. http://www.sparknotes.com/physics/optics/geom/section3.rhtml

3. http://farside.ph.utexas.edu/teaching/302l/lectures/node114.html

4. http://hyperphysics.phy-astr.gsu.edu/Hbase/hframe.html