Question

An object is placed on the left of a converging lens at a distance that is less than the focal length of the lens. The image produced will be a) real and inverted. c) virtual and inverted. b) virtual and erect. d) real and erect.

Short answer

Question: When an object is placed at a distance less than the focal length of a converging lens, the image formed will be: a) real and inverted b) virtual and erect c) virtual and inverted d) real and erect Answer: b) virtual and erect

Step-by-step solution

Review the lens formula and lensmaker's equation

The lens formula for a converging lens is given by (1/f) = (1/u) + (1/v), where f is the focal length, u is the object distance, and v is the image distance. The lensmaker's equation is given by 1/f = (n-1)((1/R1)-(1/R2)), where n is the refractive index, R1 and R2 are the radii of the lens surfaces.

Apply the lens formula with given information

Using the lens formula, we have to find v, the image distance, to determine the type of image formed. In this case, the object distance (u) is less than the focal length (f). Taking u < f, (1/f) = (1/u) + (1/v) Now, since u < f, the left side of the equation (1/f) will be smaller than the middle term (i.e., 1/u). Therefore, the term (1/v) must be negative to balance the equation. If 1/v is negative, this means that v must be negative. When the image distance (v) is negative, it indicates that the image is virtual.

Determine the orientation of the image

When a virtual image is formed using a converging lens, the image is always erect. This occurs because the light rays appear to diverge from the image point, but they never actually cross. The image is created by the extension of these diverging rays.

Choose the correct answer

Based on the analysis and calculations above, the image produced will be virtual and erect. The correct option is: b) virtual and erect.

Key concepts

Lens Formula

The lens formula is a crucial equation in optics, especially when dealing with lenses. For a converging lens, the formula is represented as \[ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} \]where:

• \( f \) is the focal length of the lens.
• \( u \) is the object distance from the lens.
• \( v \) is the image distance from the lens.

To find the image location, solve for \( v \) using the object's position \( u \) and lens's focal length \( f \).
If the object is closer to the lens than the focal length, the formula reveals different possibilities.
For instance, when \( u < f \), the calculation results in a negative \( v \), implying a virtual image formation. Virtual images occur when rays seem to converge, but don't actually meet on the real side of the lens.

Virtual Image

A virtual image is one of the fascinating outcomes when using lenses, especially converging ones. In simple terms, a virtual image cannot be projected onto a screen. This is because the image is formed at a location where the light rays appear to intersect but do not physically meet.
In our context, with a converging lens where the object is placed closer than the focal length, a virtual image is formed on the same side as the object.

• This image is created by the extension of diverging light rays backwards.
• The distance to this virtual image \( (v) \) is represented as negative in calculations, highlighting its virtual nature.

Virtual images are typically found in everyday life, such as the images we see in a magnifying glass, which showcase objects larger than their actual size.

Erect Image

When we talk about erect images, we are referring to images that have the same orientation as the object. This means the top side of the object faces the same side as the top of the image itself.
In the exercise situation utilizing a converging lens, the virtual image formed will always be erect. This happens because, while the rays suggest convergence, they never physically cross.

• This erect nature applies to virtual images formed by converging lenses with objects positioned closer than their focal length.
• Such images are not inverted, which contrasts with many real images that are flipped upside-down.

An erect image provides a direct comparison to the object, making it intuitive to understand and analyze.