Draw and use ray diagrams for the formation of a virtual image by a converging lens

3.2.3 Thin Lenses – Virtual Image Formation by a Converging Lens

Learning Objective

Draw and use ray diagrams to predict the position, nature and size of a virtual image formed by a converging (convex) lens.

Key Concepts

• A converging lens is thicker at the centre than at the edges.
• Principal axis – a straight line passing through the centre of the lens and both focal points.
• Focal point (F) – the point where parallel rays converge after refraction.
• Optical centre (O) – the geometric centre of the lens; a ray passing through O continues undeviated.
• Virtual image – formed when the refracted rays diverge; the image cannot be projected on a screen.

Ray Diagram Construction – Step by Step

1. Draw the principal axis as a horizontal line.
2. Mark the optical centre (O) of the lens on the axis.
3. Mark the focal point (F) on the side of the object (the side from which light originates). For a converging lens, the focal length \$f\$ is positive.
4. Place the object (an upright arrow) between the focal point and the lens, i.e. at a distance \$u\$ such that \$0 < u < f\$.
5. Draw the three principal rays from the top of the object:

   • Parallel ray: a ray parallel to the principal axis strikes the lens and refracts through the focal point on the opposite side.
   • Focal ray: a ray directed towards the focal point on the object's side emerges from the lens parallel to the principal axis.
   • Central ray: a ray passing through the optical centre O continues in a straight line without deviation.

6. Extend the refracted rays backward (behind the lens) until they intersect. The point of intersection is the location of the virtual image.
7. Mark the image:

   • It will be upright (same orientation as the object).
   • It will be larger than the object (magnified).
   • It will be located on the same side of the lens as the object, between the lens and the focal point.

Using the Lens Formula

The thin‑lens equation relates object distance \$u\$, image distance \$v\$, and focal length \$f\$:

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

For a virtual image formed by a converging lens, \$v\$ is negative (image on the same side as the object). Example calculation:

Rearranging the lens formula:

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

Substituting the example values:

\$v = \frac{(-6)(+10)}{-6 - 10} = \frac{-60}{-16} = +3.75\ \text{cm}\$

Because the calculated \$v\$ is positive, we interpret it as \$v = -3.75\ \text{cm}\$ using the sign convention (image on the same side as the object). Thus the virtual image forms 3.75 cm behind the lens, between the lens and the focal point.

Magnification

The linear magnification \$m\$ is given by:

\$m = \frac{h'}{h} = -\frac{v}{u}\$

where \$h'\$ is the image height and \$h\$ the object height. Using the example:

\$m = -\frac{-3.75}{-6} = -\frac{3.75}{6} = -0.625\$

The negative sign indicates an upright image, and the magnitude \$|m| = 0.625\$ shows the image is smaller than the object in this particular numerical example (if the object is placed very close to the lens, the image becomes larger). The sign convention for magnification is consistent with the ray diagram.

Common Mistakes to Avoid

• Confusing the focal point on the object side with the focal point on the image side. For a converging lens the two focal points are symmetric about the lens.
• Drawing the parallel ray through the wrong focal point. It must pass through the focal point on the opposite side of the lens.
• Forgetting to extend the refracted rays backwards to locate a virtual image.
• Applying the sign convention incorrectly in the lens formula; remember \$u\$ and \$v\$ are negative when measured from the lens towards the incoming light.

Suggested Diagram

Summary Checklist

• Identify lens type (converging) and sign of focal length (+).
• Place object between lens and focal point (\$0).
• Draw the three principal rays correctly.
• Extend refracted rays backward to find the virtual image.
• Use the lens formula with proper sign conventions to verify position.
• Calculate magnification to confirm image size and orientation.