Question

A telescope, consisting of two lenses, has an objective lens with refractive power \(0.186 \mathrm{D}\) and an eyepiece with focal length of \(5.33 \mathrm{~cm}\). What is the absolute value of the angular magnification of this telescope?

Short answer

Answer: The absolute value of the angular magnification of the telescope is approximately 100.8.

Step-by-step solution

Find focal length of the objective lens

As we know, the refractive power of a lens = 1 / focal length (in meters). Given, the refractive power of the objective lens is 0.186 D. Therefore, its focal length can be calculated as follows: Focal_length_objective = 1 / Refractive_power = 1 / 0.186 = 5.376 m

Find refractive power of the eyepiece

Now, we have the focal length of the eyepiece, which is given as 5.33 cm or 0.0533 m. We can find the refractive power of the eyepiece using the same formula as used above: Refractive_power_eyepiece = 1 / Focal_length_eyepiece = 1 / 0.0533 = 18.75 D

Calculate the angular magnification

The formula for angular magnification is the absolute value of the ratio of the refractive power of the eyepiece to the refractive power of the objective lens: Angular_magnification = | Refractive_power_eyepiece / Refractive_power_objective | Angular_magnification = | 18.75 / 0.186 | = 100.8 Hence, the absolute value of the angular magnification of the telescope is approximately 100.8.

Key concepts

Refractive Power

Understanding refractive power is essential when dealing with lenses and their applications, such as in telescopes. Refractive power, denoted by the symbol 'D', is a measure of how strongly a lens can bend light. It is inversely proportional to the focal length of the lens and is expressed in diopters (D).

The formula to calculate refractive power is simply: \[Refractive\_power = \frac{1}{Focal\_length\_(in\_meters)}\]
For an objective lens in a telescope with a refractive power of 0.186 D, the focal length is the reciprocal of this value, which plays a crucial role in determining the telescope's magnifying ability. It's important to note that a lens with higher refractive power will have a shorter focal length, indicating it is more powerful in bending light rays and vice versa.

Focal Length

Focal length, often discussed in the context of lens optics, is the distance from the center of a lens to its focal point. This distance determines how much the lens converges or diverges light. In simple terms, the focal length affects the image size. The longer the focal length, the larger the image projected onto the focal plane.

In the context of telescopes, the objective lens's focal length determines the image's size before it is magnified by the eyepiece. By manipulating the focal lengths of the lenses in a telescope, manufacturers can control the magnification power and the field of view offered by the instrument. Focal length also directly relates to the refractive power of a lens as they are inversely proportional.

Lens Optics

Lens optics is a critical area in physics, dealing with how lenses focus light. A lens can be convex (converging) or concave (diverging), with each type affecting light differently. Convex lenses are thicker in the middle and are used to focus light, while concave lenses are thinner in the middle and used to spread light out.

Application in Telescopes

In telescopes, the objective lens is typically a large convex lens that gathers and focuses light. This light is then magnified by the eyepiece lens. Both lenses work together to achieve the desired magnification and clarity in the resulting image. It is the careful arrangement and quality of these lenses that determine the performance of a telescope, highlighting the significance of lens optics in its design.

Telescope Magnification

Telescope magnification is a fundamental concept for anyone interested in astronomy or terrestrial observations. It refers to the telescope's ability to make distant objects appear larger and more detailed when viewed through the eyepiece.

The magnification of a telescope is calculated by taking the focal length of the objective lens and dividing it by the focal length of the eyepiece. However, in terms of refractive powers, the magnification formula can be represented as the absolute ratio of the eyepiece's refractive power to that of the objective lens:\[Telescope\_magnification = \left| \frac{Refractive\_power\_eyepiece}{Refractive\_power\_objective} \right|\]
This calculation gives us an 'angular magnification', which represents how much larger an object will appear compared to when it is seen with the naked eye. High magnification is desirable for viewing distant celestial objects, but it must be balanced with the telescope's ability to gather light and produce a clear image.