How are convex mirrors used?

Spherical mirrors


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This version is out of date. You can find a more up-to-date version at http://wwwex.physik.uni-ulm.de/lehre/gk3a-2003

Subsections


(See Tipler, Physik [Tip94, 1062])

Curved mirror

In the case of a curved mirror, the object becomes in the picture pictured. is the center of curvature of the mirror, therefore the angles are of incident and reflected rays equal to this line. The following applies (outside angle)

(4.1)

and (also outside angle)
(4.2)

We eliminate
(4.3)

For small angles () it holds that , and is, where the Object distance, the Image distance, the radius of curvature of the mirror and is the distance of the rays from the optical axis. Used:
(4.4)

When the object is in infinity is . We call this width the





The imaging equation, which applies not only to spherical mirrors but also to lenses, is





The above mirror is a concave mirror (French: la cave: cellar (with a vaulted cellar)). In the case of a convex mirror, the mapping equation also applies Focal length but is negative.

Spherical aberration

Rays that the paraxial approximation hurt, are not focused on one point. They form one Caustic, the image of a point is expanded. This imaging error, which is inherent in all spherical imaging devices, is called spherical aberration.

Convex mirror

(See Tipler, Physik [Tip94, 1067])
Convex mirror

The calculation of the imaging equation is analogous to that of a concave mirror. The following relations can be written down for the angles:

(4.5)

as

(4.6)

We eliminate and receive

(4.7)

The same applies to small angles , and . We get deployed

(4.8)

If we compare this equation with the equation (4.4) we see that formally the same mapping equation applies if we use the Image distance and the Focal length negative choose.

We hang on to:





Image construction with the concave mirror

(See Tipler, Physik [Tip94, 1065])

The image construction in a concave mirror is based on the following rules:

  • Each axially parallel ray runs after the reflection through the Focus (or its extension backwards goes through the Focus).
  • Every ray through that Focus becomes an axially parallel ray (or any ray whose extension by the Focus would go after the reflection to an axially parallel beam).
  • Each radial ray passes through the center of curvature of a mirror and is mapped back into itself.
  • Every ray that is directed to the vertex of the mirror (where the optical axis meets the mirror) is reflected at the same angle to the optical axis.
Image formation in the concave mirror

The illustration shows how an image is constructed according to the above rules. The fact that the rays do not cross at one point is because we do not have any paraxial rays to have.

Image scale

The Image scale is calculated by analyzing the ray for the vertex. From the theorem of rays results

(4.9)

in which the height of the item and is the height of the image.

Simplification of the construction

For paraxial rays the construction can be simplified by replacing the curved surface with a tangent plane at the vertex of the mirror. It can be seen from the image that the image that has become imprecise due to the spherical aberration becomes exact again. The newly introduced level is called Main level.

Sign conventions for mapping

G+Object in front of the mirror (real object)
 -Object behind the mirror (virtual object)
b+Picture in front of the mirror (real picture)
 -Picture behind the mirror (virtual picture)
r, f+Center of curvature in front of the mirror (concave mirror)
 -Center of curvature behind the mirror (convex mirror)
Imaged with a convex mirror

With a convex mirror, there is a virtual image behind the mirror. The illustration shows the construction of the picture.



Next page:Image generation by refraction Upwards:Geometric optics Previous page:Level mirror
This version is out of date. You can find a more up-to-date version at http://wwwex.physik.uni-ulm.de/lehre/gk3a-2003Othmar Marti
Experimental physics
Ulm University