Core Unit III: Light
B. Reflection

3. Curved Mirrors

Key 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.

A cylindrical mirror has the shape of a portion of a cylinder.

A converging mirror has a concave reflecting surface. A diverging mirror has a convex reflecting surface.

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

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

The radius of curvature (R) is the distance from the centre 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.)

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 thePrinciple 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).

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.

Image Characteristics

(These characteristics should be developed experimentally, and verified with the use of ray diagrams and equations. Rote memorization should be discouraged and avoided.)

Mirror equations

Symbols used: H_o is the height of the object, H_i is the height of the image, m is the magnification, d_o (or p) is the distance between the object and the vertex (or the distance between the object and the lens), d_i (or q) 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:

power of mirrors and lenses (in dioptres):

,where f (metre) is the focal length.

(Some texts use D for the power)

curved mirror and lens equation:

, since

(The equations apply for mirrors and lenses. The derivation of the equations using similar triangles is optional.)

Newtonian form

Symbols used: S_o is the distance between the object and the principal focus, S_i is the distance between the image and the principal focus.

(The derivations are optional.)

Either the Gaussian or Newtonian forms of the equations may be used to accommodate different approved resources, but the two systems should not be used interchangeably. The Gaussian form is preferred, but the use of reciprocals may make this form too difficult for some students to apply. Fractions can be converted to decimal form immediately to simplify calculations. So = do - f, and Si = di - f may be used, if desired, to show the equivalence between the two forms of the equations.

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. (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.

The use of equations, ray diagrams, and experimental techniques are complementary methods used to determine image characteristics in optical systems.

Learning Outcomes

Students will increase their abilities to:

1. Define the following terms: converging mirror, concave surface, diverging mirror, convex surface, vertex, principal axis, focal plane, centre of curvature, radius of curvature, focal length, paraxial rays, axial point, principal focus, spherical mirror, cylindrical mirror, aberration, spherical aberration, parabolic mirror, conjugate points.

2. Explain the Principle of Reversibility.

3. Distinguish between a concave and a convex surface.

4. Draw diagrams of converging and diverging mirrors, showing the principal axis and important points located on the principal axis for each.

5. Explain the difference between a focal point and a focal plane.

6. Explain one way that spherical aberration can be corrected in a curved mirror.

7. Express the relationship between the focal length and the radius of curvature of a curved mirror.

8. Apply the relationship between the focal length and the radius of curvature of a curved mirror in solving problems.

9. Use the rules for drawing ray diagrams for converging and diverging mirrors (parallel-ray method) to position an object on the principal axis and locate the position and other characteristics of the image.

10. Interpret the characteristics of an image from a ray diagram.

11. Demonstrate an understanding of the importance and use of a procedure of verification when using ray diagrams and equations.

12. Observe and explain that the image position in either a converging or a diverging mirror depends on the location of the object.

13. Observe and explain that except for the image position, all other characteristics of an image formed in a diverging mirror are independent of the object position.

14. Observe and explain that the characteristics of an image formed in a converging mirror depend on the object position.

15. Apply mirror equations to solving problems.

16. Apply the sign conventions for mirror equations correctly when solving problems.

17. Recognize that ray tracing and the use of equations are techniques that developed from experimentation.

Images formed by a converging spherical mirror
		Characteristics of the Image
a) Distant object		Real Inverted Smaller than object At F
b) Object beyond C		Real Inverted Smaller Between C and F
c) Object at C		Real Inverted Same size as object At C
d) Object between F and V		Virtual Erect Larger than object Behind mirror
e) Object at F		No image Reflected rays are parallel
Images formed by a diverging spherical mirror
e) Object at F		Characteristics of the image regardless of object postion Virtual Erect Smaller than object Behind mirror between F and V

Teaching Suggestions, Activities and Demonstrations

1. Perform an activity to investigate image formation in converging and diverging mirrors.

   Set up a concave mirror on the end of an optical bench. Find the image positions of an illuminated object placed at different distances from the mirror. Describe the image characteristics for all possible cases. Develop ray diagrams for each specific case.

   Determine the focal length of the mirror. (The light source will have to be placed slightly off-axis if the image is between the mirror and the illuminated object, otherwise the screen used to locate the image will block the light travelling from the object to the mirror. Another way to overcome this is to use a screen with a hole in the centre, over which is placed a sheet of clear wrap, or a fine screen.)

   Repeat with a convex mirror, describing the image characteristics, and comparing the results to those obtained with the concave mirror.

2. Develop experimentally, and verify with the use of ray diagrams or equations, the characteristics of an image formed in a converging mirror.

   An optical bench can be used to develop lens equations. This provides an excellent opportunity to reinforce practical applications of graphical and numerical analysis.

3. Explain the need for using sign conventions when solving problems using lens equations.

4. Reciprocals can be avoided when using the Gaussian form of lens and mirror equations by using the following:

   

   ,

   

   ,

   For enrichment, better students could be challenged to derive these.