LC-2: Mirrors and Lenses

OBJECTIVES:

To study image formation and focal lengths of mirrors and lenses.

APPARATUS:

Optical bench; optical components (10) kit: lenses & mirrors (5), telescope, illuminated arrow light source & 12V power supply, sharply pointed rod, desk lamp, white card or screen, plane mirror.
NOTE: the objects on the optical bench slides are not necessarily centered with respect to the index on the slide.

INTRODUCTION:

We will design, test and then measure mirror and lens assemblies by several techniques. In all instances it will be possible to use a simple interactive Java applet to perform a virtual pre-lab exercise. For the physical set-ups it will require testing location of image by absence of parallax, and others require focusing a telescope for parallel rays. For both techniques see Appendix 4. Please master this material: your instructor is available for help.

Experiment I. Radius of curvature and focal length of a concave mirror:

VIRTUAL PRE-LAB:

1.

If using the web-lab manual launch the virtual application by clicking on the Convex Mirror Application button below.

2.

After clicking on ``Start Me'' you should observe a concave mirror, an object and its image. On the lower left corner is a cursor position readout.

3.

With respect to the figure below and the mirror equation. Find the focal length.

4.

Adjust the object to get $p=q$, and obtain the focal length again.

5.

Move the object so that $p=f$. Do you see an image?

6.

Move the object so that $p<f$. Is there an image? Is it real or virtual? Is the image height smaller or larger than the object?

The Mirror Equation is given by: $$\frac{1}{p} + \frac{1}{q}=\frac{2}{R} = \frac{1}{f}$$ where $f$ is the focal length and $R$ is the mirror radius curvature. Note that when the object and image are equally distant from the mirror, $p = q = R$. You can use this condition to get an approximate value for $R$ and $f$.

Figure 1: Layout of concave lens experiment.

SUGGESTED PROCEDURE (Note: The mirror holder contains a concave and a convex mirror on opposite sides. Be sure you use the concave mirror!)

I. By using object and image distances:

1.

Resolve the image of the illuminated arrow formed by the concave mirror on the white card. Experiment with varying object distances, $p$, until you able to follow how the image distance, $q$, varies with the object distance.

2.

Measure the image distance at several different positions of the illuminated arrow object. From each pair of conjugate object and image positions calculate $f$ for the mirror. Make a table of the results in your lab book, but leave space to compare with the results of II. and III. below.

II. Obtaining $f = R/2$ by imaging at center of curvature ($q=p=R$): Replace the illuminated arrow and screen with the rod and place the rod at $R$. If the tip of the rod is at the center of curvature, a real inverted image will appear just above it with the tip and its image coinciding. An absence of parallax (see appendix D) between the tip and image is the most sensitive test for coincidence. When the parallax vanishes, the distance between the tip and the mirror is the radius of curvature $R$. From several determinations of $R$, (switching roles with your lab partners estimate the reliability of your results.)

Figure 2: Object and image at center of curvature, $R$.

III. By placing object at the principal focus ($p=f$): If the tip of the rod is at $f$, rays from the tip will reflect from the mirror as a parallel bundle and give a sharp image of the tip in a telescope focused for parallel rays (see appendix D). Keep the distance between tip and telescope small or the reflected light may miss the telescope. The best test for locating the focal point is absence of parallax between the tip's image and the cross hairs in the telescope, but the telescope must already be focused for parallel rays (e.g., focus on something out the window). From several settings estimate the reliability of your $f$ measurement.

Figure 3: Object at $f$, image at infinity.

Experiment II: LENSES

VIRTUAL PRE-LAB:

1.

If using the web-lab manual launch the virtual application by clicking on the Converging Lens Application button below.

2.

After clicking on ``Start Me'' you should observe a convex lens, an object and its image. On the lower left corner is a cursor position readout.

3.

With respect to the figure below and the lens equation. Find the focal length.

4.

Move the object so that its inverted image has the same height. Find the focal length at this point.

5.

Move the object so that $p=f$. Do you see an image?

6.

Move the object so that $p<f$. Is there an image? Is it real or virtual? Is the image height smaller or larger than the object? What is the sign of $q$?

1. Converging lens with short focal length:

From the thin lens formula

$$\frac{1}{\mbox{object distance}} + \frac{1}{\mbox{image distance}} = \frac{1}{p} + \frac{1}{q} = \frac{1}{f}$$

Measure $f$ using the three methods diagramed below (by adjustments in the object distance). Compare the results from the various methods.
I.

II.

III. OPTIONAL

2. Focal length of a diverging lens:

Since the image is always virtual it is necessary to combine it with a converging lens and configured so that the final image is real.

VIRTUAL PRE-LAB:

1.

If using the web-lab manual launch the virtual application by clicking on the Diverging Lens Application button below.

2.

After clicking on ``Start Me'' you should observe a concave lens, an object and its virtual image. On the lower left corner is a cursor position readout.

3.

With respect to lens equation and starting positions, find the focal length. Can you find an object distant in which the image height and object height are the same?

4.

Now restart the simulation with a combination of a convex and concave lens by clicking on the ``Add 2nd lens'' link.

5.

This 2nd lens will render a final image which is real. The image with a ``1'' by it is that of just the convex lens while the image with a ``2'' next to it is that of the pair. Move the object and alter its height, if necessary, so that the first image has the relationship $q_1 \approx 2 p_2$. (In this simulation $\vert f_2\vert = 3 \vert f_1\vert$ and qualitatively resembles the figure shown below.)

6.

Now repeat the last step with $\vert f_2\vert = \vert f_1\vert$ by clicking on the ``Another 2nd lens'' link. Finally reset to the ``Another 2nd lens'' and then click on the convex lens and drag it past the diverging lens.

I.

Use the set-up sketched below. First adjust Lens$_1$ so that with Lens$_2$ removed, $q_1 \sim 2p_1$. Measure $q_1$, insert the diverging Lens$_2$ reasonably close to Lens$_1$ and then locate the new image distance $q_2$. From $p_2 = d- q_1$ (a virtual object) and $q_2$ calculate $f_2$.

II.

OPTIONAL: Find $f_2$ by this more sensitive method: With the pointed rod as object now detect the image by a telescope focused for parallel rays. (As shown in the following figure.)

If using the web-lab manual, launch the virtual application by clicking on the Dual Lens Test Application button below to obverse a point source in action.

As a last step you will move the ``point'' source object (i.e. vary $p_1$ and hence $q_1$) until a sharp image appears in the telescope with no parallax relative to the cross hairs. (This occurs only if parallel rays leave the diverging lens.) The image from the converging lens at $q_1$ serves as a virtual object for the concave lens, Lens$_2$. After properly adjusting $p_1$ this virtual object will be at the focal point of the diverging lens, Lens$_2$. Obtain q$_1$ by setting p$_1$ and using the known value of $f_1$. (Make sure that $q_1$ is large enough to accommodate $d+\vert f_2\vert$). Now look for the image in the telescope and adjust $p_1$. Note that since $q_2= \infty$, $f_2 = p_2$ and $p_2 = d- q_1$.

OPTIONAL
Experiment III: Focal Length of a Convex Mirror

Once again, since convex mirrors give virtualy images it is necessary to study the mirror in combination with a converging lens.

Measure the focal length of the convex mirror by combining it with a converging lens. Set the pointed rod at twice the focal length of the lens ($p_1=2 f_1$). Next adjust the mirror position until the inverted image position shows no parallax with the object. Then $R = d-2f_1$ and

$$f_{\mbox{\small mirror}} = f_2 = R/2.$$

Of course the lens must have $f_1 > f_2$, so use a long focal length lens.