The separation between the objective and the eyepiece of a compound microscope can be adjusted between 9.8 cm to 11.8 cm. If the focal lengths of the objective and the eyepiece are 1.0 cm and 6 cm respectively, find the range of the magnifying power if the image is always needed at 24 cm from the eye.

Explanation

To find the range of the magnifying power of a compound microscope, we will follow these steps:

### Given Data:
- Focal length of the objective lens, \( F_o = 1 \, \text{cm} \)
- Focal length of the eyepiece lens, \( F_e = 6 \, \text{cm} \)
- Distance of the image from the eyepiece, \( V_e = 24 \, \text{cm} \)
- Separation between the objective and eyepiece, \( d = 9.8 \, \text{cm} \) to \( 11.8 \, \text{cm} \)

### Step 1: Calculate the object distance from the eyepiece (\( U_e \))
Using the lens formula:
\[ \frac{1}{V_e} - \frac{1}{U_e} = \frac{1}{F_e} \]
Rearranging gives:
\[ \frac{1}{U_e} = \frac{1}{V_e} + \frac{1}{F_e} \]
Substituting the known values:
\[ \frac{1}{U_e} = \frac{1}{24} + \frac{1}{6} \]
Calculating the right side:
\[ \frac{1}{U_e} = \frac{1}{24} + \frac{4}{24} = \frac{5}{24} \]
Thus,
\[ U_e = \frac{24}{5} = 4.8 \, \text{cm} \]

### Step 2: Calculate the image distance from the objective (\( V_o \))
**Case 1: Separation \( d = 9.8 \, \text{cm} \)**
\[ V_o = d - U_e = 9.8 - 4.8 = 5 \, \text{cm} \]

**Case 2: Separation \( d = 11.8 \, \text{cm} \)**
\[ V_o = d - U_e = 11.8 - 4.8 = 7 \, \text{cm} \]

### Step 3: Calculate the object distance from the objective (\( U_o \))
Using the lens formula for the objective lens:
\[ \frac{1}{V_o} - \frac{1}{U_o} = \frac{1}{F_o} \]
Rearranging gives:
\[ \frac{1}{U_o} = \frac{1}{V_o} - \frac{1}{F_o} \]

**For Case 1: \( V_o = 5 \, \text{cm} \)**
\[ \frac{1}{U_o} = \frac{1}{5} - \frac{1}{1} = \frac{1}{5} - 1 = -\frac{4}{5} \]
Thus,
\[ U_o = -\frac{5}{4} = -1.25 \, \text{cm} \]

**For Case 2: \( V_o = 7 \, \text{cm} \)**
\[ \frac{1}{U_o} = \frac{1}{7} - \frac{1}{1} = \frac{1}{7} - 1 = -\frac{6}{7} \]
Thus,
\[ U_o = -\frac{7}{6} \approx -1.17 \, \text{cm} \]

### Step 4: Calculate the magnifying power (\( M \))
The magnifying power of a compound microscope is given by:
\[ M = \frac{V_o}{U_o} \left(1 + \frac{D}{F_e}\right) \]
where \( D = 24 \, \text{cm} \).

**For Case 1:**
\[ M_1 = \frac{5}{-1.25} \left(1 + \frac{24}{6}\right) = -4 \left(1 + 4\right) = -4 \times 5 = -20 \]
Taking the magnitude, \( M_1 = 20 \).

**For Case 2:**
\[ M_2 = \frac{7}{-1.17} \left(1 + \frac{24}{6}\right) = -5.98 \left(1 + 4\right) = -5.98 \times 5 \approx -29.9 \]
Taking the magnitude, \( M_2 \approx 30 \).

### Conclusion:
The range of the magnifying power of the compound microscope is from **20 to 30**.

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