P23.  Ray Tracing and Image Formation for Spherical Mirrors

Before doing these problems, you'll need to have studied sections 26-3 and 4.  Those sections provide a guide for tracing rays for convex and concave mirrors and for doing quantitative calculations.  We'll do an example here.

Example 1.  Determine the position, size and orientation of the image of the object in the concave mirror shown to the right.

Solution:  Three principal rays (P, F, and C) have been traced from the tip of the object as per the discussion in the text.  Their intersection is the tip of the image.  The image is real (light rays actually converge there), inverted, and smaller than the object.

Taking measurements directly from the diagram (grid units of centimeters), we have:

f = 4.0 cm
d_o = 11.0 cm
d_i = 5.8 cm
h_o = 3.0 cm
h_i = -1.6 cm

Note the positive signs on object and image distances, indicating that they're both on the reflecting side of the mirror.  Note also the negative sign on the image size, indicating that it's inverted.

Let's check to see if the measured distances agree with known quantitative relationships.

We should have d_o/d_i = -h_o/h_i.  We'll calculate each ratio separately and then compare.

d_o/d_i = 11.0/5.8 = 1.9
 -h_o/h_i = -3.0/(-1.6) = 1.9

The results agree to 2 significant figures.

Now let's check the mirror equation:  1/d_o + 1/d_i = 1/f.  Again, we'll calculate each side separately and compare.

1/d_o + 1/d_i = 1/11.0 + 1/5.8 = 0.19
1/f = 1/4.0 = 0.26

The results differ by about 15%.  This can be attributed partly to error in the drawing but primarily to approximations made in deriving the mirror equation.  See page 856 for how the approximations are implemented.  They lead to the result f = ½R.  That's the way we drew the diagram above.  Let's see how close that result is.  The diagram to right is drawn correctly using the law of reflection.  The incident ray striking the mirror at point D is reflected at an angle equal to the angle of incidence.  Notice that the angles are measured with respect to the line CD, which is a radius of the circular arc of the mirror as well as a normal to the mirror's surface.  The reflected ray passes through the true focal point, f'.  The approximated focal point is f.  The error is certainly significant and can explain the discrepancy that we found above.

When doing ray tracing problems quantitatively, we don't normally take measurements from constructions such as we did above.  Instead, we simply use the two relationships:  d_o/d_i = -h_o/h_i and 1/d_o + 1/d_i = 1/f.  In the problems to follow, you'll make ray tracings to locate the image.  However, you'll use the quantitative relationships in order to obtain numerical results.

1. Begin by opening this template and printing 2 copies of it.

2. Use the template to construct ray tracings to locate the images for the problems below.

3. In order to construct the ray tracings, solve the problems as requested using the mirror equations.

4. Fax your completed work.

The problems are from Chapter 26 (both 2nd and 3rd editions):

21,22--This pair is about the same situation.  One problem is to make the ray tracing and the other to solve quantitatively.
23,24--This is another pair like 21 and 22 but with a convex mirror rather than concave.
28--Make the ray tracing first and then solve quantitatively as requested.