The behavior of a concave spherical mirror can be deduced from the way that it reflects a beam of parallel light rays. The rays are directed along the optical axis of the mirror.
As long as the diameter of the mirror is small compared to its radius of curvature, the parallel rays are all focused to a point, the focal point. For a spherical mirror, the focal point is half way between the mirror and its center of curvature.
The distance from the mirror to the focal point is called the focal length, and given the symbol, f.
When light from a real object strikes the curved mirror,
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some of the rays arrive parallel to the optic axis and are reflected through the focal point.•
some arrive by passing through the focus and are reflected parallel to the optic axis.•
the rays cross at the location of the image formed by the mirror.The location of the image is calculated from 1 1 1
DO +DI = f , where DOis the distance from the object to the mirror, DI is the distance from the image to the mirror, and f is the focal length. The height of the image, hI, is
given by h D D h
I
I O
= − O
, where hO is the height of the object. The minus sign indicates that if both Ds are positive, the image is inverted. If DO < f,
DI will be negative. This indicates that the image is virtual, located behind the mirror. The formula for the height continues to work, indicating that now the image is erect.
C
ONVEXS
PHERICALM
IRRORSThe behavior of a convex spherical mirror can be deduced from the way that it reflects a beam of parallel light rays. The rays are directed along the optical axis of the mirror.
As long as the diameter of the mirror is small compared to its radius of curvature, the parallel rays reflect as if from a focal point located behind the mirror. This mirror is said to have a virtual focus. As before, the focal length is half the radius of curvature, but this time it is negative. When light from a real object strikes the curved mirror,
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some of the rays arrive parallel to the optic axis and are reflected as if they originated at the virtual focal point.•
some of the rays arrive along a line that would cross the focal point.They are reflected before they reach the virtual focus and travel away parallel to the optic axis.
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the rays do not cross. However, lines extended back through the mirror surface do cross at the location of the image formed by the mirror.•
the image is virtual. No light passes through it.The location of the image is calculated as before, using 1 1 1 DO + DI = f , where DOis the distance from the object to the mirror, DI is the distance from the image to the mirror, and f is the focal length.
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DO is positive in our applications.•
f is negative for the convex mirror.•
DI is negative for our applications. The image is formed behind the mirror surface. It is virtual and erect.The height of the image, hI, is given by h D D h
I
I O
= − O
, where hO is the height of the object. The minus sign indicates that if both Ds are positive, the image is inverted. In practice, for the convex mirror, the sign is plus and the image is erect. If the object moves to makeDO < f , no particular change occurs for the convex mirror.
LENSES
When glass or plastic presents a curved surface to a beam of parallel light rays, the rays strike the surface at a variety of angles. As a result, they are refracted at a variety of angles.
If the surface is convex, the rays, bent toward the local normal, con-verge on a focus within the material.
In general, the light exits the other side of the glass before reaching a focus. The curvature of the second glass surface modifies the location of the focus.
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The two-surface object is a lens.•
The lens-maker’s formula for the focal length of the lens is1 1 1 1
1 2
f n
R R
=
(
−)
− .•
R1 is the radius of curvature of the first surface (the one the light hits first). R1 is positive if that surface bulges toward the incoming light.•
R2 is the radius of curvature of the second surface (the one the light hits second). R2 is positive if that surface bulges towards the incoming light.•
The focal length is the same on both sides of the lens.If the focal length of the lens is positive, it is called a converging lens.
It focuses parallel rays to a real focal point.
If the focal length is negative, the lens is called a diverging lens.
Parallel light passing into the lens diverges as though it came from a point on the “incoming” side of the lens. This lens has a virtual focus.
The location and size of the image are calculated with the same
formula that worked for the mirror: 1 1 1
DO + DI = f and h D D h
I
I O
= − O
. When the object sends light through the lens to form a real image on the other side, both the object distance and the image distance are positive.
This real image will be inverted.
As before, negative image distance means the image is virtual and erect. This means that the image is on the same side of the lens as the object.
A lens used as a magnifier is placed so that the object is just inside the focal length. The lens produces a virtual erect image farther from the lens than the object. This allows the viewer to hold the object closer to the eye, making the image on the retina larger. The view of the object is magnified.
L
ENSC
OMBINATIONSThe magnifier can also be used to magnify a real image.
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If the real image is formed by a long focal length lens from a distant object, the lens combination is a telescope. The purpose of the long focal length lens is to gather light to make a bright image for the magnifier to work on.•
If the real image is formed by a short focal length lens from a nearby object, the combination is called a microscope.If two lenses are placed side-by-side, the combination forms an image as if it were a single lens with an effective focal length given by
1 1 1
1 2
f + f = fEFFECTIVE .
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Because the inverses of the focal lengths appear in lens combinations, a new quantity called the power of the lens is defined as 1f .
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If the focal length is 1 meter, the lens is said to have a power of 1 diopter.A lens with a focal length of 20 cm has a power of 5 diopters.