A person with normal vision using a compound microscope with `f_o=8mm and f_e=2.5` cm , is able to bring an object 9 mm from objective in sharp focus.

Options

A. The minimum tube length during adjustment is 9.7 cm with image at D
B. The minimum tube length is 0.47 with angular magnification of eyepiece 10
C. The maximum tube length is 9.7 with angular magnification of eyepiece 11
D. none

Answer

To solve the problem step by step, we need to find the minimum tube length (L) and the angular magnification (M) of the compound microscope.

### Step 1: Identify the given values
- Focal length of the objective lens, \( f_o = 8 \, \text{mm} = 0.8 \, \text{cm} \)
- Focal length of the eyepiece lens, \( f_e = 2.5 \, \text{cm} \)
- Object distance from the objective lens, \( u_o = -9 \, \text{mm} = -0.9 \, \text{cm} \) (negative sign indicates that the object is on the same side as the incoming light)

### Step 2: Calculate the image distance from the objective lens (\( v_o \))
Using the lens formula:
\[ \frac{1}{f_o} = \frac{1}{v_o} - \frac{1}{u_o} \]
Substituting the known values:
\[ \frac{1}{0.8} = \frac{1}{v_o} - \frac{1}{-0.9} \]
\[ \frac{1}{0.8} = \frac{1}{v_o} + \frac{1}{0.9} \]
Finding a common denominator (0.8 and 0.9):
\[ \frac{1}{0.8} = \frac{0.9 + 0.8}{0.72} = \frac{1.7}{0.72} \]
Thus,
\[ \frac{1}{v_o} = \frac{1.7}{0.72} - \frac{1}{0.8} \]
Calculating \( v_o \):
\[ v_o \approx 7.2 \, \text{cm} \]

### Step 3: Calculate the object distance for the eyepiece lens (\( u_e \))
Using the lens formula for the eyepiece:
\[ \frac{1}{f_e} = \frac{1}{v_e} - \frac{1}{u_e} \]
Where \( v_e \) is the distance for normal vision, which is \( 25 \, \text{cm} \):
\[ \frac{1}{2.5} = \frac{1}{25} - \frac{1}{u_e} \]
Rearranging gives:
\[ \frac{1}{u_e} = \frac{1}{25} - \frac{1}{2.5} \]
Calculating \( u_e \):
\[ u_e \approx -2.27 \, \text{cm} \quad (\text{negative indicates virtual object}) \]

### Step 4: Calculate the minimum tube length (L)
The tube length \( L \) is given by:
\[ L = v_o + |u_e| \]
Substituting the values:
\[ L = 7.2 + 2.27 = 9.47 \, \text{cm} \]

### Step 5: Calculate the angular magnification (M)
The formula for angular magnification is:
\[ M = \frac{v_o}{|u_o|} \left(1 + \frac{D}{f_e}\right) \]
Where \( D = 25 \, \text{cm} \):
\[ M = \frac{7.2}{0.9} \left(1 + \frac{25}{2.5}\right) \]
Calculating:
\[ M = 8 \left(1 + 10\right) = 8 \times 11 = 88 \]

### Final Results
- Minimum tube length \( L \approx 9.47 \, \text{cm} \)
- Angular magnification \( M \approx 88 \)