Question

The diameter of Mars is 6794 km, and its minimum distance from the earth is 5.58 \(\times\) 10\(^7\)km. When Mars is at this distance, find the diameter of the image of Mars formed by a spherical, concave telescope mirror with a focal length of 1.75 m.

Short answer

The diameter of the image of Mars is approximately 0.00000000214 m.

Step-by-step solution

Understand the Problem

We need to find the diameter of the image of Mars as seen through a telescope. The problem gives us the diameter of Mars and its minimum distance from Earth, and also the focal length of the telescope. We will use these values to calculate the image diameter.

Use the Lens Formula

First, recognize that we will use the mirror formula for a concave mirror, which similar to the lens formula is \( \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \). Here, \( f \) is the focal length, \( v \) is the image distance, and \( u \) is the object distance. Convert the focal length from meters to kilometers to match the units. Thus, 1.75 m = 0.00175 km.

Calculate Object Distance

The object distance \( u \) is the distance from Mars to Earth, which is given as \( 5.58 \times 10^7 \) km. Substitute this value into the mirror formula.

Solve for Image Distance

Substitute the known values into the mirror equation: \( \frac{1}{0.00175} = \frac{1}{v} - \frac{1}{5.58 \times 10^7} \). Solve this equation for \( v \), the image distance.

Use Magnification Formula

Once \( v \) is found, use the magnification formula: \( m = \frac{h_i}{h_o} = \frac{v}{u} \), where \( h_i \) is the image height and \( h_o \) is the object height (diameter of Mars). Thus, \( h_i = h_o \times \frac{v}{u} \).

Substitute Mars' Diameter and Calculate Image Diameter

Substitute \( h_o = 6794 \) km and the computed \( v \) into the magnification formula to find \( h_i \), the image diameter. Convert this diameter to meters, since the focal length was initially given in meters.

Key concepts

Concave Mirror

A concave mirror is a type of mirror that is curved inward, resembling a portion of the interior of a sphere. These mirrors are capable of focusing light to a specific point. This unique property is what makes them useful in telescopes and other optical instruments.

The way concave mirrors work is based on the principle of reflection. When light rays parallel to the principal axis (an imaginary line that splits the mirror in half) hit the surface of a concave mirror, they reflect back through the focus. This focus is a point along the principal axis where these reflected rays meet or converge.

• Used to form real or virtual images, depending on the position of the object.
• Commonly found in devices like telescopes, headlights, and shaving mirrors.

The focal length of a concave mirror, the distance from the mirror to the focus, determines its strength to magnify or minimize an image. In the given exercise, the concave mirror in the telescope helps form an image of Mars by collecting and focusing light rays from the planet.

Image Formation

In telescope optics, image formation involves using mirrors or lenses to visually present distant objects in a perceivable format. For concave mirrors, image formation relies on the mirror formula, given as: \[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]Here, \(f\) is the focal length, \(v\) is the image distance (the distance from the mirror to where the image is formed), and \(u\) is the object distance (how far the object is from the mirror).

In the exercise, Mars is the object; thus, the formula is used to determine where the image of Mars forms once it reflects off the mirror. We must ensure units are consistent—for example, converting the focal length from meters to kilometers.

For most telescopes utilizing concave mirrors:

• The image can be real (formed on the same side as the object) or virtual (on the opposite side, behind the mirror).
• The clear, sharp image depends on accurately calculating the image distance.

This mathematical approach helps us locate where the stark and recognizable image of celestial bodies will appear through the concave mirror.

Magnification Formula

The magnification formula is a helpful tool in determining how much larger or smaller the image will appear in comparison to the actual object. With concave mirrors, the formula is:\[ m = \frac{h_i}{h_o} = \frac{v}{u} \]Where \( h_i \) is the image height, \( h_o \) is the object height, \( v \) is the image distance, and \( u \) is the object distance.

This formula states that the magnification (\( m \)) of the image is proportional to the ratio of image distance to object distance. By rearranging to find \( h_i \), we can determine the image diameter of Mars once calculated.

• If \( m > 1 \), the image is larger than the object.
• If \( m < 1 \), the image is smaller.
• If \( m = 1 \), the image is the same size as the object.

In this specific exercise, substituting the given planet diameter and distances helps calculate the diameter of Mars' image as it would appear through the telescope's concave mirror. Understanding these relationships is crucial to predicting and interpreting how objects from space can be represented visually.