The focal length of the objective and eyepiece of a telescope are 60 cm and 5 cm respectively. The telescope is focused on an object 360 cm from the objective and the final image is formed at a distance of 30 cm from the eye of the observer. Calculate the length of the telescope.

Answer

To solve the problem, we need to calculate the length of the telescope using the given parameters. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the given data
- Focal length of the objective lens, \( f_o = 60 \, \text{cm} \)
- Focal length of the eyepiece lens, \( f_e = 5 \, \text{cm} \)
- Distance of the object from the objective lens, \( u_o = -360 \, \text{cm} \) (negative as per sign convention)
- Distance of the final image from the eyepiece, \( v_e = -30 \, \text{cm} \) (negative as it is on the same side as the object for the eyepiece)

### Step 2: Use the lens formula for the objective lens
The lens formula is given by:
\[ \frac{1}{f} = \frac{1}{v} - \frac{1}{u} \]
For the objective lens:
\[ \frac{1}{f_o} = \frac{1}{v_o} - \frac{1}{u_o} \]
Substituting the known values:
\[ \frac{1}{60} = \frac{1}{v_o} - \frac{1}{-360} \]
This simplifies to:
\[ \frac{1}{60} = \frac{1}{v_o} + \frac{1}{360} \]

### Step 3: Solve for \( v_o \)
Rearranging the equation gives:
\[ \frac{1}{v_o} = \frac{1}{60} - \frac{1}{360} \]
Finding a common denominator (which is 360):
\[ \frac{1}{v_o} = \frac{6}{360} - \frac{1}{360} = \frac{5}{360} \]
Thus,
\[ v_o = \frac{360}{5} = 72 \, \text{cm} \]

### Step 4: Use the lens formula for the eyepiece
Now, we apply the lens formula for the eyepiece:
\[ \frac{1}{f_e} = \frac{1}{v_e} - \frac{1}{u_e} \]
Substituting the known values:
\[ \frac{1}{5} = \frac{1}{-30} - \frac{1}{u_e} \]
Rearranging gives:
\[ \frac{1}{u_e} = \frac{1}{-30} - \frac{1}{5} \]
Finding a common denominator (which is 30):
\[ \frac{1}{u_e} = -\frac{1}{30} - \frac{6}{30} = -\frac{7}{30} \]
Thus,
\[ u_e = -\frac{30}{7} \, \text{cm} \]

### Step 5: Calculate the length of the telescope
The length of the telescope \( L \) is given by:
\[ L = v_o + |u_e| \]
Substituting the values:
\[ L = 72 + \frac{30}{7} \]
Converting 72 into a fraction:
\[ L = \frac{504}{7} + \frac{30}{7} = \frac{534}{7} \approx 76.29 \, \text{cm \]

### Final Answer
The length of the telescope is approximately \( 76.3 \, \text{cm} \).

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