Image Formation by Lenses

Have a look at the photos below. They show two different examples of image formation by lenses One shows the image of a house, inverted and diminished. And the other shows the image of a postage stamp, enlarged and upright. You may think the lenses used are very different since the images are, but they are the same lens! If you have a magnifying glass at home, you can verify that the images formed change with the distance.

Wonder why this happens? Then keep reading. We will talk about different lenses and explain how they work. Then we will use basic rules to describe image formation by lenses.

How does an image form when using lenses?

Lenses work by using the refraction of light.

Light gets refracted when interacting with the lens because it moves through the air and the lens at different speeds. Depending on the lens's shape, an object's light can converge to a point or diverge from it, forming an image.

We can't project virtual images because the light rays of a virtual image do not converge.

One of the most important properties of an image is its magnification.

We can measure magnification using the following formula.

$\text{magnification =}\frac{\text{image height}}{\text{object height}}$

Since the magnification is a ratio it has no units.

Image formation by convex lenses

A convex lens is curved or rounded outwards.

Light rays parallel to the principal axis converge at the focus, StudySmarter Originals

Note that light refracts as it goes from the air into the lens and again as it goes back into the air. Since we can use the lens in both directions, we can identify two foci at the same distance from the lens's geometrical centre - also called the optical centre. The distance from the lens centre to its focus is called focal distance.

The focal length is the distance from the focus to the geometrical centre of the lens. StudySmarter Originals

We can understand how convex lenses form images using ray diagrams. Ray diagrams consider that light rays only refract at one point and use a simpler representation for the lens. Below is a ray diagram representing the same convex lens shown before. We can label the foci as$F_1$and$F_2$.

In a ray diagram, a convex lens is represented using a line segment with two arrow heads pointing outward on its ends. StudySmarter Originals

In general, a convergent lens is thicker in the middle.

Converging lenses are thicker in the middle than the edges, StudySmarter Originals

Rules for image formation by convex lenses

The behaviour of the light rays that go through a convex lens can be summarized as three basic rules.

1. Light rays parallel to the principal axis refract passing through the focus on the other side.
2. Light rays that go through the optical centre don't deflect.
3. Light rays passing through the focus refract parallel to the principal axis.

The behaviour of light rays going through a convex lens can be simplified by considering three special cases. StudySmarter Originals

Examples of image formation by convex lenses

We can have different types of image formation when using a convex lens. The properties of the images formed depend on the object's distance,$O$. We can distinguish five cases:

1. The object is beyond two focal distances$(O>2{\mathrm{F}}_2)$.
2. The object is exactly at two focal distances$(O=2{\mathrm{F}}_2)$.
3. The object is between one and two focal distances$({\mathrm{F}}_2<O<2{\mathrm{F}}_2)$.
4. The object at the focus$(O={\mathrm{F}}_2)$.
5. The object is between the focus and the lens$(O<{\mathrm{F}}_2)$.

Case 1: Object placed beyond two focal distances

We can find the image's position by drawing two light rays from the top of the object. The top of the image will be where these rays meet. Let's draw two light rays using rules 1 and 3.

Image formation by a convex lens for an object placed beyond two focal distances. Adapted from Kvr.lohith (CC BY-SA 4.0)

In this case, the image is:

• Real
• Diminished
• Inverted
• Formed beyond the focus but before two focal distances.

This is the same example of image formation as in the photo showing the image of a house at the beginning of the article!

Case 2: Object placed exactly at two focal distances

Let's repeat the same procedure. For this case, the image is:

Case 3: Object placed between one and two focal distances

Under these conditions, the image is:

• Real
• Inverted
• Enlarged
• Formed beyond two focal distances

Image formation by a convex lens for an object placed between F2 and 2F2. Adapted from Kvr.lohith (CC BY-SA 4.0)

Case 4: Object placed at the focus

This case is peculiar. The light rays are parallel after refracting and never intersect. Therefore, we say the image forms at infinity.

Image formation by a convex lens for an object placed at F2 on the principal axis. Adapted from Kvr.lohith (CC BY-SA 4.0)

The image formed will be:

• Real
• Inverted
• Highly enlarged
• Formed at infinity

Case 5: Object placed between the focus and the lens

In this case, the refracted rays don't intersect and move away from each other. However, if we extend the light rays backwards, they intersect behind the object. This is a different type of image formation. The light rays appear to come from behind the lens. Since the light rays do not really intersect the image is virtual.

Image formation by a convex lens for an object placed between F2 and the optical centre. Adapted from Kvr.lohith (CC BY-SA 4.0)

In this case, the image produced will be:

• Virtual and upright
• Magnified
• Behind the object

Magnifying glasses are an application of this case. That is why they can make enlarged images. This is the same example of image formation as in the photo of the stamp's image at the beginning of the article!

Correcting farsightedness with convex lenses

When we see an object, its light goes through a transparent structure in our eyes - the cornea - and then through a crystalline lens. Our eyes adjust the thickness of this lens so that incoming light rays converge exactly at the retina, where we have special cells acting as light receptors. However, specific eye issues can affect this process.

The eyes of a person with farsightedness converge the light rays of near objects behind the retina, perceiving a blurry image.

A farsightedness person sees near objects blurry as their light converges behind the retina, StudySmarter Originals

This condition can be corrected by using a converging lens which helps the eyes to converge the light rays at a shorter distance, allowing them to focus on the retina.

Convex lenses help by converging the light rays so that the eyes can form the image at the retina, StudySmarter Originals

Image formation by concave lenses

Concave lenses are hollowed out or rounded inwards. The following image illustrates how light rays passing through a concave lens disperse.

A concave lens makes the light rays diverge. StudySmarter Originals

The following ray diagram represents the same situation.

In a ray diagram, a concave lens is represented using a line segment with two arrowheads pointing inwards on its ends. StudySmarter Originals

In general, a divergent lens is thicker on its edges.

Divergent lenses can have different forms, but they are thinner in the middle than in the edges, StudySmarter Originals

Rules for Image formation by concave lenses

We can summarize the behavior of light rays as going through concave lenses as three rules.

1. Light rays parallel to the principal axis diverge appearing to come from the focus.
2. Light rays going through the optical center don't deviate.
3. Light rays going towards the focus refract moving parallel to the principal axis.

The behaviour of light rays going through a concave lens can be simplified by considering three special cases. StudySmarter Originals

Example of image formation by concave lenses

Have a look at the picture below for an object between one and two focal distances. Tracing two rays according to the previous rules we can see that light rays appear to intersect in front of the object.

Image formation by a concave lens. Kvr.lohith (CC BY-SA 4.0)

The image formed by the concave lens is:

• Virtual and upright
• Diminished
• Formed between the object and the lens

For a concave lens, the object's position does not matter. We always obtain the same type of image formation as the properties of the image are always the same.

Correcting nearsightedness with concave lenses

The eyes of a person with nearsightedness converge light rays in front of the retina, resulting in a blurry image.

A person with nearsightedness or myopia converges the light rays of distant objects in front of the retina. StudySmarter Originals

We can correct this using concave lenses. These lenses disperse the light rays so that the eyes can converge the light at the retina.

Concave lenses help by dispersing the light rays so the eyes can converge them at the retina, StudySmarters Originals