ERHS PHYSICS

Chapter 18.1b Notes

LIGHT

Curved Mirrors

Definitions and Concepts:

A curved mirror can be thought of as consisting of a very large number of small plane mirrors oriented at slightly different angles. The laws of reflection always apply, regardless of the shape or smoothness of the surface.

A spherical mirror consists of a portion of a spherical surface. Example: reflector behind a car headlight or flashlight.

A cylindrical mirror has the shape of a portion of a cylinder. Example: the curved mirrors in your Ray Box kits.

A converging mirror has a concave reflecting surface.

A diverging mirror has a convex reflecting surface.

The geometric center of the mirror is called the vertex (V). The center of a spherical reflecting surface is called the center of curvature (C).

The principal axis (PA) is a construction line drawn on a ray diagram. The principal axis passes through the vertex and the center of curvature, and is perpendicular to the focal plane.

The radius of curvature (R) is the distance from the center of curvature to the mirror.

The distance between the principal focus (F) and the vertex is called the focal length (f)

The relationship between the focal length and the radius of curvature is:

R = 2f

The principal focus (F) is a point on the principal axis on which incident rays parallel to the principal axis either converge towards, or appear to be diverging from.

The principal focus can either be real or virtual.

An axial point is a point lying on the principal axis.

Paraxial rays are rays which make very small angles with the principal axis and lie close to the axis throughout the distance between object and image.

Spherical and cylindrical mirrors do not permit all incident rays parallel to the principal axis to converge towards (or appear to have originated from) the principal focus. This is due to spherical aberration.

An aberration is an optical defect which causes a degradation in image quality.

(Theoretically, there are an infinite number of optical aberrations. Some of the more common ones are spherical and chromatic aberration, astigmatism, and coma.)

To correct for spherical aberration in mirrors, parabolic reflectors can be used. A parabolic reflector has the shape of a parabola. (Kellner-Schmidt systems or mangin mirrors also correct for spherical aberration.) Parabolic mirrors are generally more expensive and complicated to construct than spherical mirrors.

All aberrations can not be totally removed from an optical system, although optical systems can be designed to eliminate one or several types of aberrations.

The design of optical systems involves minimizing aberrations to maximize image quality.

Rules for drawing ray diagrams for converging and diverging mirrors:

(Parenthetical remarks refer specifically to diverging mirrors. Rules 1 and 2 apply to parabolic mirrors only.)

  • 1.An incident ray that is parallel to the principal axis is reflected such that it passes through the principal focus (or appears to have originated at the principal focus).
  • 2.An incident ray passing through (or heading toward) the principal focus is reflected such that it travels parallel to the principal axis.
  • 3.An incident ray passing through (or heading toward) the centre of curvature reflects back along the same path.

Rules 1 and 2 combined, and rule 3 by itself illustrate the Principle of Reversibility. If a light ray follows a particular path through an optical system, then it will follow an identical path if it travels in the opposite direction.

The rules for drawing ray diagrams can be used to determine the characteristics of an image formed by a curved mirror.

The object, represented by an arrow, is drawn to scale parallel to the mirror with its base touching the principal axis.

Important rays are drawn from the tip of the object, reflecting from the mirror according to the rules for drawing ray diagrams for curved mirrors.

The rays represent reflected light from the object, or light produced by the object.

The apparent, or real, point of convergence of the rays represents the corresponding tip of the image in the optical system.

These two points, the tip of the object and the tip of the image, form a pair of conjugate points. If the object could be placed at the location of the image, then its image would be located at the original position of the object.

Only two of the three critical rays are needed to determine the location of the image. The third ray serves as an important method of verification.

This method is called the parallel-ray method. (Oblique ray methods are not covered in this course.)

The parallel-ray method applies only to images formed by paraxial rays.

A diverging mirror always produces an erect, diminished (m< +1), virtual image, located between the vertex and the principal focus (except if the object is placed on the surface of the mirror). Convex mirrors, like the one on your outside rearview mirror on your car, are often convex.

The position of the object determines the exact location of the image in a diverging mirror. An object located near infinity forms an image at the principal focus, or on the focal plane. This holds true as well for converging mirrors.

The image characteristics found in a converging mirror depend on the location of the object. The table below summarizes the characteristics of images found in a converging mirror based on the location of the object. See the diagram page for example sketches of these conditions.

Object Location Magnification Attitude Type Position
Near Infinity
0
Inverted Real At F
Beyond C
> -1
Inverted Real Between F & C
At C
-1
Inverted Real At C
Between F and C
< -1
Inverted Real Beyond C
Between F and V
> + 1
Erect Virtual Behind the mirror
At F Undefined     Infinity

 

 

      


Mirror equations

Gaussian form

Symbols used: Ho is the height of the object, Hi is the height of the image, m is the magnification, do is the distance between the object and the vertex (or the distance between the object and the lens), di is the distance between the image and the vertex (or the distance between the image and the lens), f is the focal length.

Linear magnification :

curved mirror and lens equation:

also

Read more about these equations here.

Newtonian form

Symbols used: So is the distance between the object and the principal focus, Si is the distance between the image and the principal focus.

SiSo = f2

Sign conventions for the use of the lens equations:

1.The focal length (f) is positive for converging mirrors and lenses, and negative for diverging ones.

2.The object distance (do) is positive for real images. (The distance is negative for a virtual object.)

3.The image distance (di) is positive for all real images and negative for virtual images.

4.Heights (Ho and Hi) are positive if measured upward from the principal axis and negative if measured downward.

5.Magnification (m) is positive if the image is erect and negative when inverted.

 

These sign conventions are necessary to get correct answers when using the mirror equations. They are needed because of the different types of image characteristics found in curved mirrors under different conditions.

Glossary of Terms


Activities: use the Optics kit and ray box to examine the characteristics of converging and diverging spherical mirrors.