As an object is moved from the surface of a thin converging lens to a focal point, over what range does the image distance very?

Explanation

To solve the problem of how the image distance varies as an object is moved from the surface of a thin converging lens to its focal point, we can follow these steps:

### Step 1: Understand the Lens Formula
The lens formula is given by:
\[ \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \]
where:
- \( v \) is the image distance,
- \( u \) is the object distance (with a negative sign for real objects),
- \( f \) is the focal length of the lens.

### Step 2: Case 1 - Object at the Surface of the Lens
When the object is at the surface of the lens, we can consider the object distance \( u \) to be 0 (since the surface of the lens is our reference point). Substituting \( u = 0 \) into the lens formula:
\[ \frac{1}{v} - \frac{1}{0} = \frac{1}{f} \]
This simplifies to:
\[ \frac{1}{v} = \frac{1}{f} \]
Thus, we find:
\[ v = 0 \]
This means that the image is formed at the lens itself when the object is at the surface.

### Step 3: Case 2 - Object at the Focal Point
Next, we consider the case when the object is at the focal point of the lens. Here, the object distance \( u \) is equal to \(-f\) (the negative sign indicates that the object is on the same side as the incoming light). Substituting \( u = -f \) into the lens formula:
\[ \frac{1}{v} - \frac{1}{-f} = \frac{1}{f} \]
This simplifies to:
\[ \frac{1}{v} + \frac{1}{f} = \frac{1}{f} \]
Rearranging gives:
\[ \frac{1}{v} = 0 \]
Thus, we find:
\[ v = \infty \]
This means that when the object is at the focal point, the image is formed at infinity.

### Step 4: Conclusion
From the two cases analyzed:
- When the object is at the surface of the lens, the image distance \( v \) is 0.
- When the object is at the focal point, the image distance \( v \) approaches infinity.

Thus, we conclude that as the object moves from the surface of the lens to the focal point, the image distance varies from \( 0 \) to \( \infty \).

### Final Answer
The image distance varies from \( 0 \) to \( \infty \).

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