Four lenses of focal length ±5cm and ±200cm are available for making a telescope. To produce the largest magnification, the focal length of the eyepiece should be:
-5 cm
Understanding Telescope Magnification
The question asks us to find the focal length of the eyepiece that will produce the largest magnification for a telescope, given a set of available lenses. We have four lenses with focal lengths: +5 cm, -5 cm, +200 cm, and -200 cm.
Principle of Telescope Magnification
For a refracting telescope, the angular magnification \(M\) is primarily determined by the ratio of the focal length of the objective lens (\(f_o\)) to the focal length of the eyepiece lens (\(f_e\)). The formula for magnification is given by:
\( M \approx - \frac{f_o}{f_e} \)
The negative sign indicates an inverted image in the case of an astronomical telescope (where both objective and eyepiece are converging lenses, \(f_o > 0\), \(f_e > 0\)). For a Galilean telescope, which uses a diverging eyepiece (\(f_e < 0\)) with a converging objective (\(f_o > 0\)), the image is erect, and the magnification magnitude is \( \left| M \right| = \left| \frac{f_o}{f_e} \right| \).
To achieve the largest magnification, we need to maximize the absolute value of \(M\), which is \( \left| \frac{f_o}{f_e} \right| \). This means we need to:
- Choose an objective lens with the largest possible positive focal length (\(f_o\)).
- Choose an eyepiece lens with the smallest possible absolute value of focal length (\(|f_e|\)).
Selecting Objective and Eyepiece Lenses
The available focal lengths are +5 cm, -5 cm, +200 cm, and -200 cm.
- Positive focal lengths (+5 cm, +200 cm) correspond to converging lenses.
- Negative focal lengths (-5 cm, -200 cm) correspond to diverging lenses.
For a standard refracting telescope setup, the objective lens is a converging lens that forms a real intermediate image. The lens with the largest positive focal length is the +200 cm lens. This should be chosen as the objective lens to maximize \(f_o\).
Now, we need to choose the eyepiece from the remaining lenses (+5 cm, -5 cm, -200 cm) to minimize \(|f_e|\). The available options for the eyepiece focal length are +5 cm, -5 cm, and -200 cm.
- Option 1: Eyepiece focal length \(f_e = +5\) cm. Here \(|f_e| = 5\) cm.
- Option 2: Eyepiece focal length \(f_e = -5\) cm. Here \(|f_e| = 5\) cm.
- Option 3: Eyepiece focal length \(f_e = -200\) cm. Here \(|f_e| = 200\) cm.
Comparing the absolute values of the possible eyepiece focal lengths (\(|f_e|\)), we see that 5 cm is the smallest value. This corresponds to both the +5 cm and -5 cm lenses.
Calculating Magnification for Eyepiece Options
Using the objective lens with \(f_o = +200\) cm, let's calculate the magnitude of magnification for the eyepiece options with the smallest absolute focal length:
- If \(f_e = +5\) cm (converging eyepiece, astronomical telescope):
\( \left| M \right| = \left| \frac{f_o}{f_e} \right| = \left| \frac{+200 \, \text{cm}}{+5 \, \text{cm}} \right| = 40 \) - If \(f_e = -5\) cm (diverging eyepiece, Galilean telescope):
\( \left| M \right| = \left| \frac{f_o}{f_e} \right| = \left| \frac{+200 \, \text{cm}}{-5 \, \text{cm}} \right| = \left| -40 \right| = 40 \)
Both +5 cm and -5 cm focal length eyepieces, when paired with the +200 cm objective, yield the same magnitude of magnification, 40. However, the question asks for the focal length of the eyepiece from the given options. The options are +200 cm, -5 cm, -200 cm, +5 cm.
Comparing the magnification magnitudes for the suitable objective (+200 cm) and available eyepiece options:
| Objective \(f_o\) | Eyepiece \(f_e\) | \(|M| = \left| \frac{f_o}{f_e} \right|\) |
|---|---|---|
| +200 cm | +5 cm | \( \left| \frac{200}{5} \right| = 40 \) |
| +200 cm | -5 cm | \( \left| \frac{200}{-5} \right| = 40 \) |
| +200 cm | -200 cm | \( \left| \frac{200}{-200} \right| = 1 \) |
Using the other available positive lens (+5 cm) as objective gives much smaller magnification magnitudes (e.g., with \(f_e = +5\) cm or -5 cm, \(|M|=|\frac{5}{\pm 5}|=1\)). Diverging lenses (-5 cm, -200 cm) are typically not used as primary objectives in this type of telescope.
Thus, the maximum magnification magnitude (40) is achieved when the objective is +200 cm and the eyepiece is either +5 cm or -5 cm. Both +5 cm and -5 cm are among the options provided for the eyepiece focal length.
The question asks for the focal length of the eyepiece. Both +5 cm and -5 cm yield the maximum magnification magnitude. Looking at the provided options and standard telescope designs, both are valid eyepiece types (converging for astronomical, diverging for Galilean). Since -5 cm is listed as one of the specific choices that yields the maximum magnification magnitude alongside +5 cm, and it is one of the given options for the eyepiece, it is a valid selection.
Therefore, to produce the largest magnification using the available lenses, the objective should be +200 cm, and the eyepiece focal length should be either +5 cm or -5 cm. Both provide a magnification magnitude of 40. The option -5 cm is one of the choices presented.
Revision Table: Telescope Components
| Component | Role | Typical Focal Length Sign (Refracting) | Effect on Magnification \(|M| = \left| \frac{f_o}{f_e} \right|\) |
|---|---|---|---|
| Objective Lens (\(f_o\)) | Collects light, forms intermediate image | Positive (+) | Larger \(f_o\) means larger \(|M|\) |
| Eyepiece Lens (\(f_e\)) | Magnifies intermediate image | Positive (+) for astronomical, Negative (-) for Galilean | Smaller \(|f_e|\) means larger \(|M|\) |
Additional Information: Types of Telescopes
Telescopes are optical instruments used to view distant objects by collecting electromagnetic radiation. Refracting telescopes use lenses, while reflecting telescopes use mirrors.
- Refracting Telescopes:
- Astronomical Telescope: Uses a converging objective lens (\(f_o > 0\)) and a converging eyepiece lens (\(f_e > 0\)). Produces an inverted final image. Suitable for viewing distant stars and planets.
- Galilean Telescope: Uses a converging objective lens (\(f_o > 0\)) and a diverging eyepiece lens (\(f_e < 0\)). Produces an erect final image. Often used in binoculars and opera glasses due to the erect image, although it has a smaller field of view than astronomical telescopes.
In this problem, using a +200 cm objective and a -5 cm eyepiece would create a Galilean telescope with a magnification magnitude of 40.
Important Questions from Ray Optics and Optical Instruments
The graph correctly representing the variation of image distance v for a convex lens of focal length f versus object distance u is:
Two slits are made 0.1 mm apart, and the screen is placed 2 m away. The fringe separation when a light of wavelength 500 nm is used is:
Resolving power of a telescope can be increased by increasing:
Match List - I with List - II.
List - I List - II (A) Contracting of Eye ball (I) Myopia (B) Controls the shape of eye lens (II) Cornea (C) Elongation of eye ball (III) Ciliary Muscle (D) Control the light entering in eyes (IV) Hypermetropia Choose the correct answer from the options given below:
Which of the following statements are correct?
(A) When light rays undergo two internal reflections inside a raindrop, a secondary rainbow is formed.
(B) The angle between the emergent ray and the angle of the prism is called the angle of deviation.
(C) Light undergoes successive total internal reflections as it moves through an optical fiber.
(D) A telescope provides angular magnification of distant objects.
(E) A simple magnifier is a diverging lens of small focal length.
Choose the correct answer from the options given below:
