Question

A convex lens of focal length 10 cm, a concave lens of focal length 15 cm, and a third lens of unknown focal length are placed coaxially in contact. If the focal length of the combination is +12 cm, find the nature and focal length of the third lens, if all lenses are thin. Will the answer change if the lenses were thick?

Hint:

For thin lenses in contact, simply add the powers (\( \frac{1}{f} \)). For thick lenses, account for the separations between principal planes, which introduces additional terms in the combined focal length formula.

Answer 1

Part 1: Thin Lenses
Part 1: Thin Lenses
Step 1: Identify given values.
- Convex lens: \( f_1 = 10 \, \text{cm} \) (converging).
- Concave lens: \( f_2 = -15 \, \text{cm} \) (diverging).
- Third lens: \( f_3 \), unknown.
- Combined focal length: \( F = 12 \, \text{cm} \) (converging).

The combined focal length \( F \) of thin lenses in contact is the sum of their individual focal lengths' reciprocals:
\[\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} + \frac{1}{f_3}\]

Step 2: Substitute known values into the formula.
\[\frac{1}{12} = \frac{1}{10} + \frac{1}{-15} + \frac{1}{f_3}\]

Step 3: Calculate \( \frac{1}{f_3} \).
\[\frac{1}{12} \approx 0.0833 \, \text{cm}^{-1}\]
\[\frac{1}{10} = 0.1 \, \text{cm}^{-1}, \quad \frac{1}{-15} \approx -0.0667 \, \text{cm}^{-1}\]
\[\frac{1}{10} + \frac{1}{-15} = 0.1 - 0.0667 \approx 0.0333 \, \text{cm}^{-1}\]
\[0.0833 \approx 0.0333 + \frac{1}{f_3}\]
\[\frac{1}{f_3} = 0.0833 - 0.0333 = 0.05 \, \text{cm}^{-1}\]
\[f_3 = \frac{1}{0.05} = 20 \, \text{cm}\]

Step 4: Determine the type of the third lens.
As \( f_3 = 20 \, \text{cm} \) is positive, the third lens is a converging (convex) lens.

Step 5: Verify the result.
\[\frac{1}{F} = \frac{1}{10} + \frac{1}{-15} + \frac{1}{20} = 0.1 - 0.0667 + 0.05 \approx 0.0833\]
\[F = \frac{1}{0.0833} \approx 12 \, \text{cm}\]
The calculated combined focal length matches the given value, confirming the accuracy of \( f_3 \).


% Part 2: Thick lenses
Part 2: Effect of Thick Lenses
Thin lenses approximation assumes zero thickness, allowing direct addition of powers. For thick lenses in contact:
- Lens thickness creates separation between principal planes. Let these separations be \( d_1 \) and \( d_2 \).
- The focal length formula for two separated lenses is:
\[\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}\]
For three thick lenses, the formula is more complex, including terms like \( -\frac{d_1}{f_1 f_2} \) and \( -\frac{d_2}{f_2 f_3} \). Without lens thicknesses, an exact calculation is impossible. However, the separation would alter \( f_3 \), while the converging nature of the combination would likely persist.


% Final Answer
Final Answer:
- For thin lenses, the third lens is convex with a focal length of \( 20 \, \text{cm} \).
- For thick lenses, the effective focal length of the third lens would be modified by the separations between principal planes. The lens would likely remain converging, but its precise focal length would differ.