An astronomical telescope if ten -fold angular magnification has a length of 44 cm. The focal length of the objective is

Options

A. 440 cm
B. 44 cm
C. 40 cm
D. 4 cm

Answer

To find the focal length of the objective lens in an astronomical telescope with a given angular magnification and length, we can follow these steps:

### Step 1: Understand the relationship between magnification and focal lengths
The angular magnification (m) of an astronomical telescope is given by the formula:
\[ m = \frac{f_o}{f_e} \]
where:
- \( f_o \) = focal length of the objective lens
- \( f_e \) = focal length of the eyepiece lens

### Step 2: Write the equation for the length of the telescope
The length (L) of the telescope is the sum of the focal lengths of the objective and the eyepiece:
\[ L = f_o + f_e \]

### Step 3: Substitute known values
From the problem, we know:
- \( m = 10 \)
- \( L = 44 \, \text{cm} \)

Using the magnification formula, we can express \( f_e \) in terms of \( f_o \):
\[ f_e = \frac{f_o}{m} = \frac{f_o}{10} \]

### Step 4: Substitute \( f_e \) into the length equation
Now we can substitute \( f_e \) into the length equation:
\[ L = f_o + f_e \]
\[ 44 = f_o + \frac{f_o}{10} \]

### Step 5: Combine the terms
To combine the terms, we can express \( f_o \) in a single fraction:
\[ 44 = f_o + 0.1 f_o \]
\[ 44 = 1.1 f_o \]

### Step 6: Solve for \( f_o \)
Now, we can solve for \( f_o \):
\[ f_o = \frac{44}{1.1} \]
\[ f_o = 40 \, \text{cm} \]

### Conclusion
The focal length of the objective lens is \( 40 \, \text{cm} \).

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