In a laboratory four convex lenses `L_(1), L_(2), L_(3)` and `L_(4)` of focal lengths `2,4,6` and `8 cm` respectively are available. Two of these lenses form a telescope of length `10 cm` and magnifying power 4 . The objective and eye lenses are

Options

A. `L_(2),L_(3)`
B. `L_(1),L_(4)`
C. `L_(3),L_(2)`
D. `L_(4),L_(1)`

Answer

To solve the problem, we need to determine which of the lenses \( L_1, L_2, L_3, \) and \( L_4 \) are used as the objective and eyepiece of the telescope based on the given information about the focal lengths and the properties of the telescope.

### Step-by-Step Solution:

1. **Identify the Focal Lengths:**
- The focal lengths of the lenses are:
- \( L_1: 2 \, \text{cm} \)
- \( L_2: 4 \, \text{cm} \)
- \( L_3: 6 \, \text{cm} \)
- \( L_4: 8 \, \text{cm} \)

2. **Set Up the Equations:**
- The total length of the telescope is given as \( 10 \, \text{cm} \). This length is the sum of the focal lengths of the objective lens (\( F_o \)) and the eyepiece (\( F_e \)):
\[ F_o + F_e = 10 \quad \text{(Equation 1)} \]

3. **Use the Magnifying Power:**
- The magnifying power (\( M \)) of the telescope is given as \( 4 \). The magnifying power is defined as the ratio of the focal length of the objective lens to the focal length of the eyepiece:
\[ M = \frac{F_o}{F_e} = 4 \quad \text{(Equation 2)} \]

4. **Express \( F_o \) in terms of \( F_e \):**
- From Equation 2, we can express \( F_o \) as:
\[ F_o = 4 F_e \]

5. **Substitute \( F_o \) into Equation 1:**
- Substitute \( F_o \) from the previous step into Equation 1:
\[ 4 F_e + F_e = 10 \]
- This simplifies to:
\[ 5 F_e = 10 \]

6. **Solve for \( F_e \):**
- Dividing both sides by \( 5 \):
\[ F_e = 2 \, \text{cm} \]

7. **Find \( F_o \):**
- Now substitute \( F_e \) back into the equation for \( F_o \):
\[ F_o = 4 F_e = 4 \times 2 = 8 \, \text{cm} \]

8. **Identify the Lenses:**
- From the focal lengths given:
- \( F_e = 2 \, \text{cm} \) corresponds to \( L_1 \)
- \( F_o = 8 \, \text{cm} \) corresponds to \( L_4 \)

9. **Conclusion:**
- Therefore, the objective lens is \( L_4 \) and the eyepiece lens is \( L_1 \).

### Final Answer:
- The objective lens is \( L_4 \) (focal length 8 cm) and the eyepiece lens is \( L_1 \) (focal length 2 cm).