Problem 78 - Master Resource Book in JEE Main Physics

Question

A planet is observed by an astronomical refracting telescope having an objective of focallength \(16 \mathrm{~m}\) and an eyepiece of focal length \(2 \mathrm{~cm}\) (a) The distance between the objective and the eyepiece is \(16.02 \mathrm{~m}\) (b) The angular magnification of the planct is \(-800\) (c) The image of the planet is imverted (d) The objective is larger than the eyepiece

Step-by-Step Solution

Answer

All observations (a, b, c, d) are correct for this telescope setup.

Step 1: Understand the structure of the telescope

The telescope in question is a refracting telescope, which uses lenses as its primary means of magnification. It has two main components: the objective lens and the eyepiece lens.

Step 2: Calculate the distance between the objective and eyepiece for clear vision

The distance between the objective and the eyepiece for clear vision is the sum of their focal lengths. Given the focal length of the objective is 16 m (or 1600 cm), and the eyepiece is 2 cm,\[ \text{Total distance} = f_o + f_e = 16 ext{ m} + 0.02 ext{ m} = 16.02 ext{ m} \]Hence, the given distance of 16.02 m matches this calculation.

Step 3: Calculate the angular magnification

The angular magnification \(M\) of a refracting telescope is given by the negative ratio of the focal length of the objective \(f_o\) to the focal length of the eyepiece \(f_e\):\[ M = - \frac{f_o}{f_e} = - \frac{1600 ext{ cm}}{2 ext{ cm}} = -800 \]This indicates that the calculation provided in the exercise is correct.

Step 4: Analyze the nature of the image

The negative sign in the magnification indicates that the image is inverted. Refracting telescopes typically form inverted images, which aligns with observation \(c\) provided: "The image of the planet is inverted".

Step 5: Compare the size of the objective and the eyepiece

In refracting telescopes, the objective lens is typically larger than the eyepiece to gather more light and increase resolution. This is consistent with observation \(d\): "The objective is larger than the eyepiece."

Key Concepts

Angular Magnification

Angular magnification is a key aspect of understanding how telescopes make distant objects appear larger. In a refracting telescope, this magnification occurs mainly due to the two lenses: the objective and the eyepiece. The term refers to the ratio of the size of the angle subtended by the image to the angle subtended by the object at the eye. For refracting telescopes, the mathematical formula for angular magnification \( M \) is given by:
\[ M = - \frac{f_o}{f_e} \]where \( f_o \) is the focal length of the objective lens and \( f_e \) is the focal length of the eyepiece lens.
In the original exercise, it was calculated that the angular magnification is \(-800\). The negative sign simply indicates the image is inverted, which will be discussed further. This high magnification means that the planet being observed appears 800 times larger than it would to the naked eye. However, it's important to remember that the quality and brightness of the image are not just dependent on magnification.

• Angular magnification helps viewers see more detail.
• High magnification can sometimes lead to dimmer images if the objective lens doesn't gather enough light.

Inverted Image

An inverted image might seem like a disadvantage when using a telescope, but it is actually quite common and expected with refracting telescopes. This happens because of the way light passes through the lenses. To understand this, consider how lenses work:

• Light from a distant object first hits the objective lens, which bends (refracts) the light inward in a process called convergence.
• The converged light then passes through the eyepiece lens which further manipulates the light to make the image visible to the viewer's eye.

This bending and squeezing of light through multiple lenses results in an upside-down or inverted image. Many astronomers and telescope users are not bothered about this inverted image because:

• Objective observing (like looking at stars or planets) doesn't require the image to be upright.
• If needed, additional optics can be used to "flip" the image, although this might reduce the brightness or quality slightly.

Objective and Eyepiece Lenses

In any refracting telescope, the two main lenses are crucial for its functionality: the objective and the eyepiece. Each has a specific purpose, contributing to the magnification and clarity of the image seen through the telescope.

• Objective Lens: This is the larger lens at the front of the telescope. Its primary role is to gather as much light as possible from the distant object, like a planet or a star. The more light it collects, the better the resolution and brightness of the final image.
• Eyepiece Lens: This smaller lens is located closer to your eye when you look through the telescope. It takes the light converged by the objective and spreads it out for your eye to interpret as a magnified image.

In the original exercise, the focal length of the objective is significantly larger than that of the eyepiece (16 m vs. 0.02 m). This setup is what allows the telescope to achieve its powerful magnification of \(-800\).
Choosing the right combination of objective and eyepiece determines the telescope’s effectiveness at observing distant bodies. A larger objective lens is particularly important for:

• Gathering more light, leading to brighter images.
• Increasing resolution, allowing finer details to be seen.