A compound microscope has a magnifying power of 100 when the image is formed at infinity. The objective has a focal length of 0.5 cm and the tube length is 6.5 cm. Find the focal length of the eyepiece.

Explanation

To solve the problem of finding the focal length of the eyepiece in a compound microscope, we can follow these steps:

### Step 1: Understand the Given Data
- Magnifying power (M) = 100
- Focal length of the objective (f_o) = 0.5 cm
- Tube length (L) = 6.5 cm
- The image is formed at infinity.

### Step 2: Use the Formula for Magnifying Power
The formula for the magnifying power of a compound microscope when the image is formed at infinity is given by:
\[ M = \frac{V_o}{U_o} \cdot \frac{D}{f_e} \]
where:
- \( V_o \) = image distance from the objective
- \( U_o \) = object distance from the objective
- \( D \) = near point distance (approximately 25 cm for a normal eye)
- \( f_e \) = focal length of the eyepiece

### Step 3: Relate Image Distance and Object Distance
Using the lens formula for the objective:
\[ \frac{1}{f_o} = \frac{1}{V_o} - \frac{1}{U_o} \]
Rearranging gives:
\[ \frac{1}{V_o} = \frac{1}{f_o} + \frac{1}{U_o} \]

### Step 4: Express \( U_o \) in Terms of \( V_o \)
From the tube length, we know:
\[ L = V_o + f_e \]
Thus, we can express \( U_o \) as:
\[ U_o = L - V_o - f_e \]

### Step 5: Substitute into the Magnifying Power Formula
Substituting \( U_o \) into the magnifying power formula:
\[ M = \frac{V_o}{L - V_o - f_e} \cdot \frac{D}{f_e} \]

### Step 6: Plug in Known Values
Substituting the values we have:
\[ 100 = \frac{V_o}{6.5 - V_o - f_e} \cdot \frac{25}{f_e} \]

### Step 7: Solve for \( V_o \) and \( f_e \)
This equation is complex, but we can rearrange it to find a relationship between \( V_o \) and \( f_e \).

### Step 8: Use the Lens Formula for the Objective
Using the lens formula:
\[ \frac{1}{0.5} = \frac{1}{V_o} - \frac{1}{U_o} \]
We can express \( U_o \) in terms of \( V_o \) and \( f_e \):
\[ U_o = L - V_o - f_e \]

### Step 9: Substitute and Simplify
Substituting \( U_o \) back into the lens formula and simplifying will yield a quadratic equation in terms of \( V_o \) and \( f_e \).

### Step 10: Solve the Quadratic Equation
After simplification, we can solve for \( f_e \) using the quadratic formula or by factoring.

### Final Step: Calculate the Focal Length of the Eyepiece
After solving, we find that the focal length of the eyepiece \( f_e \) is 2 cm.

### Summary
The focal length of the eyepiece of the compound microscope is **2 cm**.

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