# Reflection at a Spherical Surface. Concave Mirrors

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Reflection at a Spherical Surface Slide 2 Concave Mirrors Slide 3 The center of curvature of the surface is at point c Point p is the Object p v Point v is the vertex of the mirror The line pcv is called the optic axis. It is sometimes referred to as the principle axis. Slide 4 Concave Mirrors Slide 5 p p is known as the real image o p is known as the object p Slide 6 Concave Mirrors p ooo p s s There exists a relationship between the distances s and s relative to the curvature or the reflecting surface, R. That Relationship is 1 1 2 ------- + ------- = ------ s s R R Slide 7 Paraxial Approximations The trigonometric calculations for spherical mirrors in determining focal points, radii and such (as in the last slide) are based on accepting that the tangent of small angles are nearly equal to the angle itself (in radians) and that rays are assumed parallel to the optic axis at distances far from the mirror. The term paraxial approximations is used when referring to these calculations. Slide 8 Planar Spherical Mirrors p ooo p s s As R approaches infinity, the mirror becomes a plan mirror and the above equation reduces to s = -s 1 1 2 ------- + ------- = ------ s s R R Slide 9 spherical aberration p ooo p s s As increases point p moves closer to the vertex reducing the sharpness of the image. this is known as spherical aberration. 1 1 2 ------- + ------- = ------ s s R Slide 10 When the center of curvature C is on the same side as the outgoing (reflected) light, the radius of curvature is positive: otherwise it is negative. R will always be positive for a concave mirror and negative for a convex mirror ooo Sign Rules... No. 3 Slide 11 Lateral Magnification (spherical mirror) p ooo p s s The formula for Lateral Magnification is: y M = -------- y R The ratio of the height of the object to the height of the real image is know as lateral magnification ( y vs. y ) y y If m is negative the image is inverted, relative to the object Slide 12 Lateral Magnification and Distance (spherical mirror) p ooo p s s This relationship is: y - y -------- = ------- s s Since the two triangles are similar, there exists a proportionality between the magnification and the distance. y y Therefore: y - s m = -------- = ------- y s Again, If m is negative the image is inverted, relative to the object Slide 13 Lateral magnification in a spherical mirror Can you mathematically determine the magnification in a planar mirror ? s = - s and y = y y - s -s m = -------- = ------- = ------- = 1 y s -s Slide 14 If an object is very far away from a spherical mirror ( s = ) what is the formula for the radius of the mirror in terms of the image distance (s) ? 1 1 2 R --- + --- = --- s = ---- s R 2 Slide 15 Given: A concave mirror forms an image, on a wall 3 m from the mirror, of a filament of a headlight lamp 10 cm in front of the mirror. (a) what is the radius of curvature of the mirror? (b) what is the height of the image if the height of the object is 5 mm? (a) both object distance and image distance are positive. s = 10 cm and s = 300 cm 1 / 10 cm + 1 / 300 cm = 2 / R R = 19.4 cm (b) m = y / y = - s / s = -300 cm / 10 cm = -30 The image is therefore inverted (m is negative) and is 30 times the height of the object, or 30 x 5 mm = 150 mm Slide 16 Focal Point The point F at which the incident parallel rays converge is called the Focal Point The distance from the vertex to the focal point F is called the Focal Length F The focal length (f) is related to R by f = R/2 Slide 17 The focal length (f) is related to the distance by: 1 1 1 --- + --- = --- s s f (object-image relation spherical mirror) Slide 18 Do problems 34.3 and 34.5

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