Thin Lenses

3.2.3 Real Images by Converging Lenses

A converging lens can form real images under certain conditions. A real image is formed when light rays actually converge to a point.

Ray Diagram Construction

To draw a ray diagram for a converging lens forming a real image, we use three principal rays:

1. The Parallel Ray: A ray of light traveling parallel to the principal axis is refracted through the focal point (F) of the lens.
2. The Centre Ray: A ray of light passing through the centre of the lens continues in a straight line.
3. The Focal Ray: A ray of light passing through the focal point (F) of the lens is refracted parallel to the principal axis.

To find the location of the real image, we draw these three rays and mark the point where they intersect.

Image Characteristics

The characteristics of the real image formed by a converging lens depend on the position of the object relative to the focal point (F):

• Object Distance (u): The distance of the object from the lens.
• Image Distance (v): The distance of the image from the lens.
• Image Type:
  • If the object is beyond 2F, the image is inverted and real.
  • If the object is between F and 2F, the image is inverted and real.
  • If the object is between the lens and F, the image is upright and virtual. (This is formed by diverging lenses)
• Image Size (Magnification): The ratio of the image height to the object height. $$m = \frac{h'}{h}$$ where:
  • $h'$ is the height of the image.
  • $h$ is the height of the object.
  The magnification can be positive (upright image) or negative (inverted image).

Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of the lens:

$$ \frac{1}{f} = \frac{1}{u} + \frac{1}{v} $$

Where:

• $f$ is the focal length of the lens (in cm).
• $u$ is the object distance (in cm).
• $v$ is the image distance (in cm).

By rearranging the lens formula, we can find the image distance (v) if we know the object distance (u) and focal length (f):

$$ v = \frac{fu}{u - f} $$

Example Problem

An object is 20 cm from a converging lens of focal length 15 cm. Calculate the image distance and magnification.

Using the lens formula:

$$ v = \frac{(15 \text{ cm})(20 \text{ cm})}{20 \text{ cm} - 15 \text{ cm}} = \frac{300}{5} \text{ cm} = 60 \text{ cm} $$

The image distance is 60 cm.

To calculate the magnification:

Since the image distance is positive, the image is real and inverted.

$m = \frac{h'}{h} = \frac{v}{u} = \frac{60 \text{ cm}}{20 \text{ cm}} = 3$

The magnification is 3, meaning the image is 3 times the size of the object and inverted.

Table Summary

Ray	Description
Parallel Ray	Refracts through the focal point (F).
Centre Ray	Continues in a straight line.
Focal Ray	Refracts parallel to the principal axis.