The final image in an astronomical telescope adjustment, a straingt black line of length L is drawn on the objective lens. The eyepiece forms a real image of this line. The length of this image is l. The magnification of the telescope is

Options

A. virtual and erect
B. real and erect
C. real and inverted
D. virtual and inverted

Answer

To find the magnification of the astronomical telescope, we will follow these steps:

### Step 1: Understand the Components of the Telescope
An astronomical telescope consists of two lenses: the objective lens (with a larger focal length, \( F_0 \)) and the eyepiece lens (with a smaller focal length, \( F_e \)). The objective lens forms a real image of a distant object, which is then viewed through the eyepiece.

**Hint:** Remember that the objective lens focuses light to form an image, while the eyepiece magnifies that image for viewing.

### Step 2: Define the Lengths
Let \( L \) be the length of the straight black line drawn on the objective lens, and \( l \) be the length of the image formed by the eyepiece.

**Hint:** Identify the lengths involved: \( L \) is the object length, and \( l \) is the image length.

### Step 3: Write the Magnification Formula
The total magnification \( M \) of the telescope can be expressed as:
\[ M = \frac{F_0}{F_e} \]
where \( F_0 \) is the focal length of the objective lens and \( F_e \) is the focal length of the eyepiece.

**Hint:** Recall that magnification is the ratio of the image size to the object size.

### Step 4: Relate the Image and Object Lengths
The magnification can also be expressed in terms of the lengths of the object and image:
\[ M = \frac{l}{L} \]
This means that the magnification is the ratio of the length of the image \( l \) to the length of the object \( L \).

**Hint:** This relationship shows how much larger the image appears compared to the original object.

### Step 5: Equate the Two Expressions for Magnification
Since both expressions represent the same magnification, we can set them equal to each other:
\[ \frac{F_0}{F_e} = \frac{l}{L} \]

**Hint:** This step combines the two perspectives of magnification into one equation.

### Step 6: Solve for Magnification
From the equation \( \frac{F_0}{F_e} = \frac{l}{L} \), we can express the magnification \( M \) as:
\[ M = \frac{l}{L} \]

**Hint:** This final expression shows that the magnification of the telescope is directly related to the lengths of the image and the object.

### Final Answer
Thus, the magnification of the telescope is given by:
\[ M = \frac{l}{L} \]