Applications of Spherical Mirrors

 

Carlyle Moore

 

Example 3.7

 

            Equation (1.15) is known as the Gaussian form of the mirror equation.  Show that, for a concave mirror, it may be written as , where F is the focal length of the mirror, and  are the x-coordinates of the object and the image, respectively, with respect to the principal focus as origin.  This result is known as Newton's formula.

______

 

           

The coordinates  are related to the coordinates u and v by

(2.1)

 

Let us assume that the light is incident form the left.  This means that, for a concave (converging) mirror,  the principal focus is on the left (  ), and the mirror formula (1.15) may therefore may be written in the form

 

(2.2)

 

or

(2.3)

 

whence

(2.4)

 

Note that, since  is positive, the coordinates  must be of the same sign.  In other words, an object and its image formed by a concave (converging) mirror must be on the same side of the principal focus.   Moreover, Equation  (2.4) may be written in the form

 

(2.5)

 

where  are the distances of the object and the image, respectively, from the principal focus.  This means that as  increases,  decreases, and vice versa, i.e. an object and its  image in a concave mirror  move in opposite directions.

 

Example 3.8

            Obtain an expression for the lateral magnification m produced by a concave mirror as a function of the distance U of the object from the mirror. Plot a graph of m versus U and use it to show that an erect image (of an erect object) in a concave mirror is always magnified.

            Let us assume that the object is on the left of the mirror.  Then, using the Cartesian convention, we have

 

(3.1)

 

and the mirror formula (1.15) becomes

 

(3.2)

 

or

(3.3)

 

i.e.

(3.4)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


where .    A great deal of information can be obtained from a graph of m versus x.  As x increases from zero, the magnification increases steadily from , approaching  as x approaches 1 from below.  As x  decreases from m decreases from zero, approaching  as x approaches 1 from above.  In other words, if the object is further away from the mirror than the principal focus, the image may be magnified or diminished, but will always be inverted. The image is erect (  ) only if , in which case , i.e. the image is magnified.  This makes a concave mirror particularly suitable for use as a shaving mirror (or make-up mirror).  The focal length of the mirror may be chosen  so as to produce a magnification of about 1.5, when the user's face is a suitable distance (say 25 cm) away from the mirror. Observe that when the magnification  is positive, v must be positive (image on the right), since u is negative. This means that the reflected rays diverge, and the image is virtual, when .   By the same token, when the magnification is negative, v must also be negative (image on the left).  Thus the reflected rays converge, giving rise to a real image, when .  To sum up, when the object distance U is less than the focal length F of a concave mirror, the image is virtual, erect and magnified.  When U is greater than F, the image is real, inverted, and may be either magnified (when , i.e. when the object lies between the principal focus and the center of curvature) or diminished (when  ).

 

            The singularity at  arises from the fact that all paraxial rays emanating from the principal focus are reflected  parallel to the axis, i.e. the image is located at infinity.    

 

            Note, incidentally, that Equation (3.3) above may be written as

 

(3.5)

 

where we have used the transformation .  In view of the Newton formula , this may now be written as

 

(3.6)

 

Example 3.9

            A concave mirror is to be used as a shaving (or make-up ) mirror which produces a lateral magnification of 1.5 when the user's face is 25 cm away from the mirror.  What must be the focal length F of the mirror?

______

 

            If the object is to the left of the mirror, we have (using the Cartesian convention)

 

(4.1)

 

and the mirror formula becomes

(4.2)

 

The lateral magnification is given by

 

(4.3)

 

Combining (4.2) and (4.3), we get

 

(4.4)

 

and the focal length of the mirror is given by

 

(4.5)

 

Aliter

 

            Alternatively, we may write the lateral magnification as

 

(4.6)

 

 

where F, and hence , is positive (O lies to the right of the principal focus.  Hence

   

(4.7)

 

(4.8)

 

            Let us now turn our attention to convex mirrors.  The general method of analysis (ray tracing, use of the mirror formulae) has been developed  with reference to concave mirrors.  A similar analysis, applied to convex mirrors, yields essentially the same results, and will not be presented again.  The only difference is that when a parallel beam of light is incident (in the paraxial region) on a convex mirror, the reflected

 

 

 

rays diverge - they appear to come from a single point G (also called the principal focus) on the axis.  A convex mirror is therefore a diverging mirror. 

 

Example 3.10

            Obtain an expression for the lateral magnification m produced by a convex mirror in terms of the distance U of the object from the mirror.  What information can be obtained from the graph of m versus U?

_____

 

            The magnification produced by a convex mirror may be found from the mirror equations (1.15) and (1.17).  We assume, as usual, that the object is on the left of the mirror,

 

(5.1)

 

Equation (1.15) then becomes

 

(5.2)

 

and the lateral magnification is

 

(5.3)

 

or

(5.4)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


where .  Recall that U and F are both distances, i.e. inherently positive quantities.  Thus it follows from Equation (5.3) that the image coordinate v is always positive, i.e. the image is located on the right of the mirror (on the opposite side from the object). The reflected rays must therefore diverge, which indicates that the image is virtual. It is also clear from Equation (5.4) that, since , the magnification m obeys the inequality .  It  follows, then, that the image is always upright (right side up) and diminished.