Question

In normal adjustment, for a refracting telescope, the distance between the objective and eyepiece is 30 cm. The focal length of the objective, when the angular magnification of the telescope is 2, will be:

• A. 20 cm
• B. 30 cm
• C. 10 cm
• D. 15 cm

Hint:

In an astronomical telescope, the focal length of the objective lens is usually larger than the eyepiece lens to provide higher magnification.

Answer 1

The Correct Option is A

Step 1: {Understanding the Normal Adjustment Condition}
In a refracting telescope under normal adjustment, the total length of the telescope is: \[ L = f_o + f_e \] where:
\( f_o \) is the focal length of the objective lens,
\( f_e \) is the focal length of the eyepiece lens.
Step 2: {Using the Given Values}
It is given that the total length of the telescope is: \[ f_o + f_e = 30 \] Also, the magnification of the telescope is given by: \[ M = \frac{f_o}{f_e} \] Since \( M = 2 \), we get: \[ \frac{f_o}{f_e} = 2 \] Step 3: {Solving for \( f_o \) and \( f_e \)}
Rewriting the equation: \[ f_o = 2 f_e \] Substituting into the length equation: \[ 2f_e + f_e = 30 \] \[ 3f_e = 30 \] \[ f_e = 10 { cm}, \quad f_o = 20 { cm} \] Thus, the correct answer is \( 20 \) cm.

Answer 2

Step 1: Understanding Normal Adjustment in Refracting Telescope
In normal adjustment, the final image is formed at infinity. In this case, the distance between the objective lens and the eyepiece lens is the sum of their focal lengths:
\[ L = f_o + f_e \] Where:
- \( L = 30 \, \text{cm} \) (given)
- \( f_o \) = focal length of objective
- \( f_e \) = focal length of eyepiece

Step 2: Angular Magnification Formula
The angular magnification \( M \) of a telescope in normal adjustment is given by:
\[ M = \frac{f_o}{f_e} \] Given: \( M = 2 \)

Step 3: Substituting Values
\[ \frac{f_o}{f_e} = 2 \quad \Rightarrow \quad f_o = 2f_e \] Now use this in the total length formula:
\[ f_o + f_e = 30 \Rightarrow 2f_e + f_e = 30 \Rightarrow 3f_e = 30 \Rightarrow f_e = 10 \, \text{cm} \] \[ \therefore f_o = 2 \times 10 = 20 \, \text{cm} \]

Step 4: Final Answer
\[ \boxed{f_o = 20 \, \text{cm}} \] So the correct answer is: Option 1: 20 cm