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Convex mirror |
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(1) Ray tracing:
Trace (at least) two rays (any direction as long as it hits the mirror) originating from the same point on the object (e.g. the head of the object).
Use the Law of reflection to obtain the reflected rays. The two reflected rays will diverge, but if you trace the rays all the way behind the mirror, the rays will intersect at one point, we will call these virtual rays (i.e. not real). This is the location of the image (image of the head of the oject). Now do the same for the tail of the object.
Note: Virtual iamge - In reality there are no light rays behind the mirror. But our brains are so use to the fact that light travels in a straight line that our brain interprets the reflected rays as coming from the back of the mirror. Remembering this idea is crucial in image construction.
Note: Images formed by plane mirrors are sharp - You can draw 3 or 4 or more rays, in fact all rays originated from one point will interesect at one point after reflection. This means that there is an unique position for the image => a sharp image. This situation is not true for spherical mirrors, there is no unique position and the image is fussy, see discussion on spherical aberration below.
(2) Algbraic method: See spherical mirror in next section.
The type of image where no actual light hitting on it is called a virtual image (only virtual rays hitting it).
An image where light hits it is called a real image (this happens for concave mirror).
We can show that for a plane mirror the image size is the same as the object size by simple geometry, we say that the magnification (m) is 1.
The image is oriented the same way as the object, we say that the image is upright.
We can combine the magnification and orientation information together and say the magnification is +1 => same size and upright. If the magnification is -1 +> same size but inverted.
Object distance (s)=distance of object from mirror.
Image distance (s') = distance of image from mirror. By convention, if the image is behind the mirror then s' is negative.
The mirror surface is a portion of a shpere => it has a radius of curvature (r) and the center (C) of the shpere is located in front of the mirror.
We called the line passing throught the center to the middle of the mirror the axis of mirror.
The angles are measured with respect to the "normal" direction to the surface. This normal direction is along a line joining the center (C) to the point of incidence on the surface. For example: Light rays eminate from the center will be reflected back to the center (what is the angle of incidence in this case?)
Now you can ray trace, but soon you will find that not all reflected rays originate from one point will intersect at one point (see Fig. 31-10, P. 1016), only rays near the axis of of mirror do so, this condition is called spherical aberration (spherical aberration can be removed by using parabollic mirrors but they are more difficult to manufacture)
To minimize shperical aberration, we use only the portion of the mirror near the axis. How close to the axis is close enough? What length scale should you compare it with?
There is a focal point (F) located in front of the mirror at a distance r/2 from the mirror. All rays parallel to the axis will be reflected toward the focal point (see Fig. 31-12 (a), P. 1017.) provided that the parallel rays are not "too far" from the axis of the mirror (remember why?)
What about light rays eminated from the focal point, how are they reflected? (This is the principle used by search lights)
(1) Ray tracing: Same method as for plane mirror; you can pick any two rays from the same point on the object. However, there are two particular rays that are most convenient:
One is the horizontal ray (i.e. parallel to the axis), it is reflected to the focal point.
One is a ray passing througth the center, it is reflected back toward the center.
(To check the accuracy of your ray tracing, it is recommended that you draw a third ray, a ray passing through the focal point will be reflected back horizontally. This third reflected ray should intersect at the same point as the previous two rays.)
Note for concave mirror, it is possible that the image is formed in front of the mirror (i.e. light rays actually hits that spot), this type of image is called real image. At what object distances, will you have a real image?
There are 3 distinct regions:
Region (1) : object distance (s) < f, region (2): f < s < r, and region (3): r < s
Construct the images for these three situation and characterize the image by stating the location of image (which region), virtual or real; enlarged or reduced; upright or inverted.
(2) Algebraic method: 1/s + 1/s' = 1/f ; f = r/2 (Eq. 31-6, P. 101)
Calculate s' for s in the three regions and compare the results with the ray tracing results (pick some numbers for s and f and calculate s')
The text shows that the magnification is related to the ratio of s and s', m=-s'/s (Eq. 31-7, P.1020.
(Remember m = positive means upright image and m=negative means inverted image).
For a single mirror, s is always positive because the object must be in front of the mirror (see footnote on P.1019 for situation where s is negative). For virtual images (behind the mirror) s' =negative, therefore m = positive => virtual images are upright. For real images, s' = positive and m =negative => real images are inverted.
Manipulate Eq. 31-6 to find s' and m (m=-s'/s). Plot s' vs s and m vs. s to summarize all the results.
Eq. 31-6 => 1/s + 1/s' =0. Does it correspond to the plane mirror result? (Calculate s' and m)
The mirror surface is a portion of a shpere => it has a radius of curvature (r) and the center (C) of the shpere is located behind of the mirror.
Same as for concave mirror, the angles are measured with respect to the "normal" direction to the surface. This normal direction is along a line joining the center (C) to the point of incidence on the surface. For example: Light rays hitting the surface along a line that traces toward the center will reflect back where it came from (again angle of incidence = 0).
There is a focal point (F) located behind the mirror at a distance - r/2 from the mirror. All rays parallel to the axis will be refected away but the virtual rays will focus at the focal point (see Fig. 31-13, P. 1017.) (Again, provided that the parallel rays are not "too far" from the axis of the mirror).
(1) Ray tracing: Same method as for concave mirror.
Again consider 3 regions:
Region (1) : object distance (s) < |f|, region (2): |f| < s < r, and region (3): r < s
Construct the images for these three situation and characterize the image by stating the location of image (which region), virtual or real; enlarge or reduced; upright or inverted. You will discover that for a convex mirror, all three region behave the same way.
(2) Algebraic method: 1/s + 1/s' = 1/f ; f = - r/2 (Eq. 31-6, P. 101)
Calculate s' for s in the three region and compare with the ray tracing results (pick some numbers for s and f and calculate s')
Manipulate Eq. 30-6 to find s' and m (m=-s'/s) and therefore show that:
For a single convex mirror, all images are virtual, inverted, and reduced.
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