A compound microscope consists of an objective of focal length 1 cm and an eyepiece of focal length 5 cm. An object is placed at a distance of 0.5 cm from the objective. What should be the separation between the lenses so that the microscope projects an inverted real image of the object on a screen 30 cm behind the eyepiece?

Explanation

To solve the problem, we need to find the separation between the objective lens and the eyepiece of the compound microscope. We will use the lens formula and the given parameters step by step.

### Step 1: Identify the given values
- Focal length of the objective lens (FO) = 1 cm
- Focal length of the eyepiece (FE) = 5 cm
- Object distance from the objective lens (UO) = -0.5 cm (negative because the object is on the same side as the incoming light)
- Image distance from the eyepiece (VE) = -30 cm (negative because the image is formed 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:
\[ \frac{1}{FO} = \frac{1}{VO} - \frac{1}{UO} \]
Substituting the known values:
\[ \frac{1}{1} = \frac{1}{VO} - \frac{1}{-0.5} \]
This simplifies to:
\[ 1 = \frac{1}{VO} + 2 \]
Rearranging gives:
\[ \frac{1}{VO} = 1 - 2 = -1 \]
Thus:
\[ VO = -1 \text{ cm} \]

### Step 3: Use the lens formula for the eyepiece
For the eyepiece, we again use the lens formula:
\[ \frac{1}{FE} = \frac{1}{VE} - \frac{1}{UE} \]
Substituting the known values:
\[ \frac{1}{5} = \frac{1}{-30} - \frac{1}{UE} \]
Rearranging gives:
\[ \frac{1}{UE} = \frac{1}{-30} - \frac{1}{5} \]
Finding a common denominator (which is 30):
\[ \frac{1}{UE} = -\frac{1}{30} + \frac{6}{30} = \frac{5}{30} \]
Thus:
\[ UE = 6 \text{ cm} \]

### Step 4: Calculate the separation between the lenses
The separation \( L \) between the objective lens and the eyepiece is given by:
\[ L = VO + UE \]
Substituting the values we found:
\[ L = (-1) + 6 = 5 \text{ cm} \]

### Final Answer
The separation between the lenses should be **5 cm**.

---