Problem 92 - Physics

Question

An astronomical telescope has an angular magnification of \(-132 .\) Its objective has a refractive power of 1.50 diopters. What is the refractive power of its eyepiece?

Step-by-Step Solution

Answer

The refractive power of the eyepiece is 198 diopters.

Step 1: Understand Angular Magnification Formula

The angular magnification of a telescope is given by the formula \( M = - \frac{F_o}{F_e} \), where \( M \) is the angular magnification, \( F_o \) is the focal length of the objective lens, and \( F_e \) is the focal length of the eyepiece. The negative sign indicates that the image is inverted.

Step 2: Relate Refractive Power to Focal Length

Refractive power (in diopters) is the reciprocal of the focal length in meters, i.e., \( F = \frac{1}{f} \). Therefore, the focal length can be found as \( f = \frac{1}{F} \). The objective's refractive power is \(1.50\) diopters, giving it a focal length \( f_o = \frac{1}{1.50} \) meters.

Step 3: Calculate Focal Length of Eyepiece

Using the angular magnification formula, substitute \( M = -132 \) and \( F_o = \frac{1}{1.50} \). Solve for the focal length of the eyepiece \( f_e \): \[ -132 = - \frac{\frac{1}{1.50}}{f_e} \] Rearranging gives: \[ 132 \cdot f_e = \frac{1}{1.50} \]\[ f_e = \frac{1}{132 \cdot 1.50} \]

Step 4: Calculate Refractive Power of Eyepiece

Find the refractive power of the eyepiece using \( F_e = \frac{1}{f_e} \). Substitute the value of \( f_e \) from the previous step:\[ F_e = \frac{1}{\frac{1}{132 \cdot 1.50}} = 132 \cdot 1.50 \] \( F_e = 198 \, \text{diopters} \).

Key Concepts

Angular Magnification

Angular magnification is a crucial concept when discussing optical instruments like telescopes. In the case of an astronomical telescope, the angular magnification (\(M\)) tells us how much larger (or smaller) an object appears when viewed through the telescope compared to the naked eye.

• The formula for angular magnification in telescopes is \(M = -\frac{F_o}{F_e}\).
• In this formula, \(F_o\) represents the focal length of the objective lens, and \(F_e\) represents the focal length of the eyepiece.
• The negative sign indicates that the image is inverted, which is typical for telescopes.

Using this relationship, we can determine either the focal length of one lens or the magnification, given the other two values. In practical terms, high angular magnification means the observed celestial objects will appear significantly larger, making it easier to study them in detail.

Refractive Power

Refractive power is another foundational concept in optics, especially relevant when dealing with lenses. It describes how strongly a lens converges (focuses) or diverges light. The refractive power (\(F\)) is measured in diopters and is determined by the reciprocal of the focal length in meters:

• \(F = \frac{1}{f}\), where \(f\) is the focal length in meters.

For an astronomical telescope, the objective lens typically has a low refractive power, meaning it has a longer focal length, suited for gathering and focusing light from distant objects. Conversely, the eyepiece, with higher refractive power, allows the viewer to comfortably observe these magnified images. Calculating the refractive power of the eyepiece involves knowing its focal length and using the relationship between these two quantities.

Focal Length

The focal length of a lens is a distance from the lens where parallel rays of light converge to a point. It is a critical factor in determining both the magnification and refractive power of lenses used in telescopes. The focal length influences:

• The field of view and the amount of detail that can be seen.
• The overall length of the telescope system, as it adds the objective and eyepiece focal lengths.
• The clarity of the image — longer focal lengths generally produce clearer images.

In a telescope system, the objective lens typically has a longer focal length, which allows it to gather more light from distant objects. The focal length of the eyepiece lens is shorter to provide the needed magnification for viewing. Calculations involving focal lengths use reciprocal relationships with refractive power, providing a way to interconvert between these two important optical parameters.