A camera with a focal length of 0.0500m (a 50mm lens) is focused for an object at infinity. To focus the camera on a subject which is 4.00m away, how should the lens be moved ?

Options

A. 1.0cm closer to the film
B. 0.06cm farther from the film
C. 0.06 cm closer to the film
D. 4.94 cm farther from the film

Answer

To solve the problem of how much the lens should be moved to focus on a subject that is 4.00 m away, we can follow these steps:

### Step 1: Understand the Lens Formula
The lens formula relates the object distance (u), the image distance (v), and the focal length (f) of the lens. The formula is given by:

\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]

### Step 2: Identify the Given Values
From the problem, we know:
- Focal length, \( f = 0.0500 \, \text{m} \) (which is 50 mm)
- Object distance for the subject, \( u = -4.00 \, \text{m} \) (the negative sign indicates that the object is on the same side as the incoming light)

### Step 3: Substitute the Values into the Lens Formula
We need to find the image distance \( v \). Rearranging the lens formula gives:

\[ \frac{1}{v} = \frac{1}{f} - \frac{1}{u} \]

Substituting the values:

\[ \frac{1}{v} = \frac{1}{0.0500} - \frac{1}{-4.00} \]

### Step 4: Calculate Each Term
Calculating \( \frac{1}{f} \):

\[ \frac{1}{0.0500} = 20 \, \text{m}^{-1} \]

Calculating \( \frac{1}{u} \):

\[ \frac{1}{-4.00} = -0.25 \, \text{m}^{-1} \]

### Step 5: Combine the Terms
Now, substituting these values back into the equation:

\[ \frac{1}{v} = 20 - (-0.25) = 20 + 0.25 = 20.25 \, \text{m}^{-1} \]

### Step 6: Calculate \( v \)
Now, take the reciprocal to find \( v \):

\[ v = \frac{1}{20.25} \approx 0.0494 \, \text{m} \]

### Step 7: Determine the Movement of the Lens
Initially, when the camera was focused on an object at infinity, the image distance \( v \) was equal to the focal length \( f = 0.0500 \, \text{m} \). Now, we have calculated \( v \approx 0.0494 \, \text{m} \).

To find out how much the lens should be moved, we calculate the difference:

\[ \text{Movement} = f - v = 0.0500 - 0.0494 = 0.0006 \, \text{m} = 0.6 \, \text{mm} \]

### Conclusion
The lens should be moved approximately **0.6 mm** closer to the image sensor to focus on the subject which is 4.00 m away.

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