Lens Combinations Laboratory Worksheet
Objective
To obtain a lens combination with a specified focal length by using two lenses from the given set of lenses.
Theory
When two thin lenses with focal lengths $f_1$ and $f_2$ are placed in contact with each other, the effective focal length $F$ of the combination is given by:
$$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$$
This is known as the lens maker's formula for a combination of lenses. The power of a lens ($P$) in diopters is defined as the reciprocal of the focal length in meters:
$$P = \frac{1}{f}$$
For a combination of lenses, the powers are additive:
$$P_{total} = P_1 + P_2 = \frac{1}{f_1} + \frac{1}{f_2}$$
A lens with a positive focal length is a converging lens, which causes parallel rays to converge to a focus. A lens with a negative focal length is a diverging lens, which causes parallel rays to diverge as if coming from a virtual focus.
When two lenses are used in combination, the resulting system can be treated as a single lens with an effective focal length. The formula $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$ allows us to calculate this effective focal length.
For example, if we combine a convex lens with focal length $f_1 = +20$ cm with a concave lens with focal length $f_2 = -40$ cm, the effective focal length would be:
$$\frac{1}{F} = \frac{1}{20} + \frac{1}{-40} = 0.05 - 0.025 = 0.025$$
$$F = \frac{1}{0.025} = 40 \text{ cm}$$
Thus, the combination would act as a converging lens with focal length 40 cm.
Materials Required
- Set of convex lenses (positive focal lengths)
- Set of concave lenses (negative focal lengths)
- Optical bench
- Lens holders
- Screen
- Light source
- Meter scale
Apparatus
Figure 1: Experimental setup for determining the focal length of lens combinations
Available Lenses
The following lenses are available for your experiment:
| Lens No. | Type | Focal Length (cm) | Power (diopters) |
|---|---|---|---|
| 1 | Convex | +10 | +10.0 |
| 2 | Convex | +15 | +6.67 |
| 3 | Convex | +20 | +5.0 |
| 4 | Concave | -15 | -6.67 |
| 5 | Concave | -20 | -5.0 |
| 6 | Concave | -30 | -3.33 |
Procedure
Set up the optical bench with the light source at one end and the screen at the other end.
Select two lenses from the available set. Record their focal lengths and powers.
Place the lenses in contact with each other in the lens holder.
Position the lens combination on the optical bench between the light source and screen.
Adjust the position of the lens combination until a sharp image of the light source is formed on the screen.
Measure the distance between the lens combination and the screen. This gives the experimental value of the focal length of the lens combination.
Calculate the theoretical focal length using the lens maker's formula and compare with the experimental value.
Repeat the experiment with different lens combinations to achieve various specified focal lengths.
Interactive Lens Combination Calculator
Select two lenses from the available set to calculate the effective focal length of the combination:
Lens 1
f = +10 cm
P = +10.0 D
f = +15 cm
P = +6.67 D
f = +20 cm
P = +5.0 D
f = -15 cm
P = -6.67 D
f = -20 cm
P = -5.0 D
f = -30 cm
P = -3.33 D
Lens 2
f = +10 cm
P = +10.0 D
f = +15 cm
P = +6.67 D
f = +20 cm
P = +5.0 D
f = -15 cm
P = -6.67 D
f = -20 cm
P = -5.0 D
f = -30 cm
P = -3.33 D
Combination Results
Select two lenses above to see the results of their combination.
Target Focal Lengths
Try to achieve the following target focal lengths using combinations of two lenses from the available set:
| Target No. | Target Focal Length (cm) | Possible Lens Combination | Theoretical Result (cm) |
|---|---|---|---|
| 1 | +6 | ||
| 2 | +30 | ||
| 3 | -10 | ||
| 4 | +60 |
Data Collection
Record your measurements for each lens combination:
| Combination | Lens 1 (cm) | Lens 2 (cm) | Theoretical Focal Length (cm) | Measured Focal Length (cm) | Percentage Error (%) |
|---|---|---|---|---|---|
| 1 | |||||
| 2 | |||||
| 3 | |||||
| 4 |
Calculations
For each lens combination, calculate the theoretical focal length using the formula:
$$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$$
Calculate the percentage error using:
$$\text{Percentage Error} = \left| \frac{F_{theoretical} - F_{measured}}{F_{theoretical}} \right| \times 100\%$$
Sample Calculation
For a combination of lenses with focal lengths $f_1 = +15$ cm and $f_2 = -30$ cm:
Theoretical focal length:
$$\frac{1}{F} = \frac{1}{15} + \frac{1}{-30} = 0.0667 - 0.0333 = 0.0334$$
$$F = \frac{1}{0.0334} = 29.94 \approx 30 \text{ cm}$$
If the measured focal length is 31.2 cm, the percentage error would be:
$$\text{Percentage Error} = \left| \frac{29.94 - 31.2}{29.94} \right| \times 100\% = 4.21\%$$
Discussion Questions
- What happens to the effective focal length when two converging lenses are combined?
- What happens to the effective focal length when a converging lens and a diverging lens are combined?
- Under what conditions would a combination of a converging and diverging lens result in a diverging lens?
- How does the power of the lens combination relate to the powers of the individual lenses?
- What are some potential sources of error in this experiment?
Conclusion
Summarize your findings and compare the theoretical and experimental values of the focal lengths for different lens combinations. Discuss any discrepancies and their possible causes.
Important Notes
- Make sure the lenses are clean before using them.
- When placing lenses in contact, ensure their optical centers are aligned.
- For accurate results, ensure the optical bench is properly aligned.
- When measuring the focal length, ensure the image formed on the screen is sharp.
- Be careful when handling glass lenses to avoid breakage or scratches.