Problem 67 - Physics for Scientists and Engineers with Modern Physics

Question

(II) The Moon's image appears to be magnified \(120 \times\) by a reflecting astronomical telescope with an eyepiece having a focal length of 3.1 \(\mathrm{cm} .\) What are the focal length and radius a of curvature of the main (objective) mirror?

Step-by-Step Solution

Answer

Focal length is 372 cm; radius of curvature is 744 cm.

Step 1: Understand the Magnification Formula

The angular magnification (or magnifying power) of a telescope is given by the formula \( M = \frac{f_o}{f_e} \), where \( f_o \) is the focal length of the objective lens or mirror, and \( f_e \) is the focal length of the eyepiece. We are provided with \( M = 120 \) and \( f_e = 3.1 \) cm.

Step 2: Rearrange the Formula to Solve for Objective Focal Length

Rearrange the magnification formula to find the focal length of the objective lens (or mirror): \( f_o = M \times f_e \). Substitute \( M = 120 \) and \( f_e = 3.1 \) cm into the equation.

Step 3: Calculate the Focal Length of the Objective Mirror

Substitute the given values into the equation: \( f_o = 120 \times 3.1 \). This gives \( f_o = 372 \) cm. Thus, the focal length of the objective mirror is 372 cm.

Step 4: Calculate the Radius of Curvature of the Objective Mirror

For a mirror, the relationship between the radius of curvature (R) and the focal length (f_o) is given by \( R = 2f_o \). Since \( f_o = 372 \) cm, substitute this into the equation: \( R = 2 \times 372 \).

Step 5: Final Calculation of Radius of Curvature

Calculate the radius of curvature: \( R = 744 \) cm. So, the radius of curvature of the main (objective) mirror is 744 cm.

Key Concepts

Focal Length

In the world of optics, the focal length holds a special significance. It essentially defines how strongly a lens or mirror converges or diverges light. To understand this better, picture the focal length as the distance between the lens or mirror and the point where light rays meet to form a clear image. This point is called the focal point.

When it comes to telescopes, the focal length of the objective mirror determines how it collects light and focuses it to form a clear celestial image. The longer the focal length, the greater the magnification potential of the telescope, assuming the eyepiece focal length remains the same.

The formula to determine the focal length of an objective is given by rearranging the magnification formula:

• Magnification formula: \(M = \frac{f_o}{f_e}\)
• Objective focal length: \(f_o = M \times f_e\)

Given the magnifying power and the focal length of the eyepiece in our original exercise (120 and 3.1 cm respectively), the focal length of the objective is calculated to be 372 cm.

This longer focal length allows the telescope to show distant objects like the Moon, magnified greatly.

Angular Magnification

Angular magnification is a concept that might sound complex but is truly about how large a viewed object appears through an optical instrument compared to the naked eye. When using a telescope, angular magnification tells us by how much the telescope is enlarging the view of the distant object.

The formula for angular magnification is crucial in calculations. It compares the focal lengths of the objective and the eyepiece:

• \(M = \frac{f_o}{f_e}\)

Here, \(M\) is the angular magnification, \(f_o\) is the focal length of the objective, and \(f_e\) is the focal length of the eyepiece. A magnification of \(120\times\), as seen in our telescope problem, means that the Moon looks 120 times larger than it does with the naked eye, all thanks to both the objective mirror's and the eyepiece's optical properties.

Keen astronomy enthusiasts adjust telescopes’ magnification by changing their eyepiece, thus altering how reels in the wonders of the universe become.

Radius of Curvature

The radius of curvature of a mirror is a fundamental concept that describes the geometry of the mirror's surface. It tells us how "curvy" the mirror is. Imagining the mirror as a part of a large imaginary sphere, the radius of curvature is simply the radius of that sphere.

For a spherical mirror, whether it's concave or convex, the radius of curvature,

• is linked to the focal length by the simple relation: \(R = 2f_o\)

In our telescope example, the focal length of the objective mirror is given as 372 cm.

Thus, the radius of curvature becomes \(744\) cm, indicating a large spherical size. This relationship is pivotal when designing telescopes, ensuring light can focus precisely to capture clear images of astronomical objects.

Understanding the radius of curvature means gaining insights into how well a mirror can direct incoming light into a localized region, enhancing both focus and clarity.