An object is placed at different distances, u, from a converging lens of focal length, 15.0cm. For what value of u does the lens act as a microscope?

Note: option C is printed as "M < 15cm only", which looks like a typo for "u < 15cm only". I'll treat it that way.

Analyze the question

A converging lens acts as a simple microscope (magnifying glass) when it produces an image that is enlarged, upright, and virtual. We need the range of object distances u that gives this.

Concept

For a converging lens with focal length f = 15.0 cm, the image depends on where the object is placed:

• u > 2f (u > 30 cm): real, inverted, diminished image (like a camera)
• u = 2f (u = 30 cm): real, inverted, same size image
• f < u < 2f (15 to 30 cm): real, inverted, magnified image (like a projector)
• u = f (u = 15 cm): image at infinity
• u < f (u < 15 cm): virtual, upright, magnified image (magnifying glass)

Solution

Use the lens formula: 1/f = 1/u + 1/v

So 1/v = 1/f − 1/u = (u − f)/(uf)

• If u < f, then (u − f) is negative, so v is negative. This means the image is virtual, on the same side as the object.
• The magnification is m = |v/u| = f/(f − u), which is greater than 1 whenever u < f. So the image is magnified and upright.

For example, if u = 10 cm: 1/v = 1/15 − 1/10 = −1/30, so v = −30 cm. The magnification is 3, and the image is virtual, upright and enlarged.

So the lens acts as a simple microscope only when u < 15 cm.

Answer

Option C (u < 15 cm only).

Tip

Remember "inside the focal length = magnifying glass." Whenever the object is closer to the lens than f, you get a virtual, upright, enlarged image. Cameras use u > 2f, and projectors use f < u < 2f.