Curved mirror
From Wikipedia, the free encyclopedia
A curved mirror is a mirror with a curved reflective surface, which may be either convex (bulging outward) or concave (bulging inward). Most curved mirrors have surfaces that are shaped like part of a sphere, but other shapes are sometimes used in optical devices. The most common non-spherical type are parabolic reflectors.
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[edit] Convex mirror
A collimated (parallel) beam of light diverges (spreads out) after reflection from a convex mirror, since the normal to the surface differs with each spot on the mirror.
[edit] Image
The image is always virtual (rays haven't actually passed though the image), diminished (smaller), and upright . These features make convex mirrors very useful: everything appears smaller in the mirror, so they cover a wider field of view than a normal plane mirror does as the image is "compressed". The passenger-side mirror on a car is typically a convex mirror. In some countries, these are labelled with the safety warning "Objects in mirror are closer than they appear", to warn the driver of the convex mirror's distorting effects on distance perception.
[edit] Concave mirrors
A concave mirror, or converging mirror, has a reflecting surface that bulges inward (away from the incident light). Unlike convex mirrors, concave mirrors show different types of image depending on the distance between the object and the mirror itself.
These mirrors are called "converging" because they tend to collect light that falls on them, refocusing parallel incoming rays toward a focus. This is because the light is reflected at different angles, since the normal to the surface differs with each spot on the mirror.
[edit] Image
Note: S here stands for distance between object and mirror.
- When <math>S<F</math>, the image is:
- Virtual
- Upright
- Magnified (larger)
- When <math>S=F</math>, the image is formed at infinity.
- Note that the reflected light rays are parallel and do not meet the others. In this way, no image is formed or more properly the image is formed at infinity.
- When <math>F<S<2F</math>, the image is:
- Real
- Inverted (vertically)
- Magnified (larger)
- When <math>S=2F</math>, the image is:
- Real
- Inverted (vertically)
- Same size
- When <math>S>2F</math>, the image is:
- Real
- Inverted (vertically)
- Diminished (smaller)
[edit] Mirror shape
Most curved mirrors have a spherical profile. These are the simplest to make, and it is the best shape for general-purpose use. Spherical mirrors, however, suffer from spherical aberration. Parallel rays reflected from such mirrors do not focus to a single point. For parallel rays, such as those coming from a very distant object, a parabolic reflector can do a better job. Such a mirror can focus incoming parallel rays to a much smaller spot than a spherical mirror can.
- See also: Toroidal reflector
[edit] Analysis
[edit] Mirror equation and magnification
The mirror equation relates the object distance (<math>d_o</math>) and image distances (<math>d_i</math>) to the focal length (<math>f</math>):
- <math>\frac{1}{d_o}+ \frac{1}{d_i} = \frac{1}{f}</math>.
The magnification of a mirror is defined as the height of the image divided by the height of the object:
- <math>m \equiv -\frac{h_i}{h_o} = - \frac{d_i}{d_o}</math>.
The negative sign in front of this fraction is used as a convention. By convention, if the magnification is positive, the image is upright. If the magnification is negative, the image is inverted (upside down).
[edit] Ray tracing
The image location and size can also be found by graphical ray tracing, as illustrated in the figures above. A ray drawn from the top of the object to the surface vertex (where the optical axis meets the mirror) will form an angle with that axis. The reflected ray has the same angle to the axis, but is below it (See Specular reflection).
A second ray can be drawn from the top of the object passing through the focal point and reflecting off the mirror at a point somewhere below the optical axis. Such a ray will be reflected from the mirror as a ray parallel to the optical axis. The point at which the two rays described above meet is the image point corresponding to the top of the object. Its distance from the axis defines the height of the image, and its location along the axis is the image location. The mirror equation and magnification equation can be derived geometrically by considering these two rays.
[edit] Ray transfer matrix of spherical mirrors
- Further information: Ray transfer matrix analysis
The mathematical treatment is done under the paraxial approximation, meaning that the under first approximation a spherical mirror is a parabolic reflector. The ray matrix of a spherical mirror is shown here for the concave reflecting surface of a spherical mirror. The <math>C</math> element of the matrix is <math>-\frac{1}{f}</math>, where <math>f</math> is the focal point of the optical device.
Boxes 1 and 3 feature summing the angles of a triangle and comparing to π radians (or 180°). Box 2 shows the Maclaurin series of <math>\arccos\left(-\frac{r}{R}\right)</math> up to order 1. The derivations of the ray matrices of a convex spherical mirror and a thin lens are very similar.
[edit] External links
- Java applets to explore ray tracing for curved mirrorsde:Hohlspiegel
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