# Physics Tutorial: Power of Lenses. The Human Eye

In this Physics tutorial, you will learn:

• What is power of lens and where it is used?
• How can we calculate power of a lens?
• How can we calculate the power of s system of lenses?
• What is the structure of human eye?
• How does the human eye work?
• What are some defects of vision?
• What can we do to correct the sight when needed?
• How can we know what kind of optical glasses are needed in various situation?
• What is the range of normal vision?

## Introduction to Power of Lenses. The Human Eye

How do people see the objects around them?

In weather forecasts there is an option called visibility. Do you think it depends by the ability of humans to see objects around them or by weather conditions? Why?

What do people use to see better when they have sight problems? Why?

Are all optical glasses identical? What do you known about them?

In this tutorial, we will focus on the use of lenses in sight correction. For this, we also need to explain the human eye as a natural optical tool. Hence, some elements contained in biophysics involving nerves, eye-parts, etc. are discusses as well. Lenses, Equation of Lenses and Image Formation of Lenses

## Power of Lenses

As we have seen in the previous tutorial, lenses are not all identical even when they belong to the same category. Their radius of curvature determines the point in which parallel rays coming from infinity converge. We called this point as focus, F and numerically, it was equal to half-distance from the lens to the centre of curvature (half of the radius).

There is a new quantity used to determine how able is a lens to collect rays at points near it. This quantity is the reciprocal of focal length and is known as (optical) Power of a lens, P. The only condition is that focal length must be written in metres. Its formula therefore is:

P = 1/F (in metres)

Power of lens is measured in Dioptre [D]. For example, a converging lens of focal length equal to 20 cm (0.2 m) has a power of

P = 1/F
= 1/0.2 m
= 5 dioptres

On the other hand, a diverging lens of focal length equal to 40 cm (0.4 m), has a power of

P = 1/F
= - 1/0.4 m
= - 2.5 dipotres

because the focal length in diverging lenses is taken as negative.

## Power of a Combination of Lenses

When two lenses are in contact, the focal length of this system is calculated by the formula

1/F = 1/F1 + 1/F2

(Here, we do not consider the lenses' thickness, which in reality makes them slightly displaced from each other.)

Therefore, the optical power of such a system of lenses, is

P = P1+ P2

### Example 1

What is the power of the system composed by two lenses as those shown in the figure, if they are placed in contact with each other?  ### Solution 1

The given values represent the distance between the centres of curvatures in each lens. Therefore, we must divide them by 4 to obtain the respective focal lengths. Thus, we have for the converging lens

F1 = d1/4
= 48 cm/4
= 12 cm

and for the diverging lens

F2 = d2/4
= 32 cm/4
= 8 cm

We take the focal length of the diverging lens as negative. Thus, we write F2 = - 8 cm.

Since these two lenses are placed in contact, we obtain for the total optical power of this system:

P = P1 + P2
= 1/F1 + 1/F2
= 1/0.12 m + -1/0.08 m
= 2/0.24 m - 3/0.24 m
= - 1/0.24 m
≈ - 4.17 dipotres

When two lenses are not in contact, we must also consider the distance between them in order to calculate the optical power of this system. The formula of optical power in this case becomes

P = P1 + P2 - d × (P1 × P2)

where d is the distance (in metres) between the two lenses and P1 and P2 are the individual optical powers.

### Example 2

Two lenses are placed at the positions shown in the figure. What is the optical power of this system of lenses?

### Solution 2

First, let's work out the respective focal lengths. From the figure, you can see that the focal length of the converging lens is F1 = 20 cm / 2 = 10 cm = 0.1 m and for the diverging lens, the focal length is F2 = 12 cm / 2 = 6 cm = 0.06 m (we must write - 0.06 m as it is the focal length of a converging lens).

Also, we can determine the distance between the two lenses. It is d = 10 cm + 8 cm + 6 cm = 24 cm = 0.24 m. Thus, the optical power of this system is

P = P1 + P2 - d × (P1 × P2 )
= 1/F1 + 1/F2 - d × (1/F1 × 1/F2 )
= 1/0.1 m + -1/0.06 m - 0.24 m × (1/0.1 m × -1/0.06 m)
= 3 - 5/0.3 - 0.24 × (-1/0.006)
= -2/0.3 + 0.24/0.006
= 20/3 + 240/6
= -6.67 + 40
= 33.33 dioptres

## The Human Eye

The human eye is a wonderful natural structure that helps us see the world in a colourful way. It is made up by a number of components, each of them having a specific function. Let's take a closer look to this important organ of our body.

### a) Parts of the eye

The eye has a spherical shape called eyeball, most of which has a white colour when healthy. The cornea is the clear front surface of the eye. It lies directly in front of the iris and pupil, and it allows light to enter the eye. Attached to the pupil there is a converging lens, which refracts the light rays entering the eye, forming an inverted image on the retina, at the back of the eyeball.

The retina has about 130 million light-sensitive cells! These cells produce electrical signals when light falls upon them. The optic nerves carry these signals from the eye to the brain. The brain then converts the signals into upright image. Although the image on the retina is inverted, we don't see the world upside down!

Ciliary muscle is a circular muscle that enables the lens to change shape for focusing.

The inside of the eyeball is filled with a watery liquid as shown in the figure below. ### b) Night and day

The light intensity entering the eye is controlled by the iris. If there is too much light, the iris enlarges, making the pupil smaller. In this way, the light intensity entering the eye decreases. This occurs during the day, in places exposed to sunlight. During the night the reverse process does occur, i.e. the iris shrinks, making the pupil larger. As a result, more light enters the eye and therefore, the light intensity increases.

### c) The focusing eye

The eye lens is flexible, i.e. it becomes thicker or thinner as needed. This is made possible by the action of ciliary nerves as stated earlier. When the eye lens changes thickness, its focal length also changes. Thus, the eye can make focusing adjustments for objects at various distances. This property of eye is known as eye accommodation. Practically, a person with normal vision has an infinite distance of eye accommodation because he/she may see objects at very high distances without having any trouble. Therefore, the far point of normal people is at infinity. On the other hand, the nearest point in which the eye can accommodate effortlessly is known as the near point. For normal people the near point is 25 cm. This means the eye can be damaged when we try to see objects closer than 25 cm, because this causes tension in the eyes.

### d) Eye defects

Many people have trouble in seeing clearly the objects around them. Their vision is blurred when it should not be so. This phenomenon occurs due to many factors, both inherited and acquired, and it is known as eye defect.

Basically, there are six types of eye defects:

#### 1. Long-sightedness (Hyperopia)

Some people see distant objects clearly but they cannot see nearby objects with the same clarity. This occurs either because their eyeballs are shorter than normal, or because their eye-lenses are too thin. Such people are said to be long-sighted. In this pathology, light rays entering patients' eyes from a near object are focused behind the retina. A converging lens placed before the eye (in the form of optical glasses) can correct this defect. It changes the direction of incident light to the eye by bringing them closer to each other, so that the eye-lens forms an image exactly on the retina, as shown in the figure below. #### 2. Near-sightedness (Myopia)

Some people see nearby objects very clearly but they cannot see distant objects with the same degree of clarity. These people are said to be near-sighted. Their eyeballs are either longer than normal, or their eye-lenses are too thick. As a result, the light rays from a distant object are focused in a position that is located before the retina. A diverging lens in the form of optical glass is used to correct this kind of defect. This lens diverges the light rays so that the eye-lens forms an image exactly on the retina, as shown in the figure below. ##### Example 3

The typical distance between the eye-lens and retina is 2.5 cm. A patient goes to an oculist who discovers that the light rays converge at a distance of 2.8 cm from the eye-lens, as shown in the figure. 1. What type of eye-defect does the patient have?
2. What type of optical glasses must the doctor prescribe to him? (Include numerical values in your answer). Suppose the optical glasses must be at 1 cm distance from the eye-lenses.
##### Solution 3
1. The patient is long-sighted because the rays converge beyond the retina. Therefore, his eye-defect is the hyperopia.
2. The doctor must prescribe optical glasses containing converging lenses, in order to make the light rays converge at a shorter distance from the eye lens, i.e. make them converge on the retina.

We must use the formula of combined lenses power in order to find the focal length of the converging lens needed (and therefore, the power of the diverging lens to be used in the optical glasses prescribed by the doctor). We have the following clues:

Feye-lens = F2 = 2.8 cm = 0.028 m
Fcombined = F = 2.5 cm = 0.025 m
d = 1 cm = 0.01 m
Flens = F1 = ?
P = P1 + P2 - d × (P1 × P2 )
1/F = 1/F1 + 1/F2 - d × (1/F1 × 1/F2 )

Substituting the known values, we obtain

1/0.025 = 1/F1 + 1/0.028 - 0.01 × (1/F1 × 1/0.028)
1000/25 = 1/F1 + 1000/28 - 10/28 × F1
40 - 1000/28 = 28/28 × F1 - 10/28 × F1
1120/28 - 1000/28 = 18/28 × F1
120/28 = 18/28 × F1
120 = 18/F1

In this way, we obtain for the focal length of the optical glasses needed:

F1 = 18/120
= 0.15 m
= 15 cm

This means the doctor must prescribe optical glasses of power

P = 1/F1
= 1/0.15 m
= +6.67 dioptres

to the patient, in order to correct his vision.

#### 3. Colour-blindness

This is another anomaly of the eye, in which the patient is not able to distinguish the colours of objects. The cells on normal peoples' retina are sensitive to colours. As a result, the objects appear coloured. However, a colour-blind person cannot recognize some colours. A colour-blind person is not able to make distinction between red and green - both of them look the same to him/her!

Try to read the hidden number in the figure. If you are able to read the number 12, then you are not colour-blind.

Colour-blind people must avoid working in jobs where the colours are important such as in painting, fashion industry, etc.

#### 4. Astigmatism

This pathology is a result of an irregularly shaped cornea or eye-lens. If the eye-lens or cornea are not perfectly spherical, the eye can form a correct image only in some directions, but not in others. To correct this defect, cylindrical lenses are typically used.

The following figure shows a simple test for astigmatism. If the figure looks regular, i.e. the distance between the lines looks the same, then your vision is regular, otherwise you should visit a doctor to see if you have any astigmatism symptoms. #### 5. Cataract

Cataract occurs when the vision becomes cloudy. Such a pathology can be corrected through a surgical intervention.

#### 6. Blindness

Blindness represents a pathology involving a lack of vision. It may also refer to a loss of vision that cannot be corrected with glasses or contact lenses. Partial blindness means the patient has very limited vision. Complete blindness means the patient cannot see anything and is not able to see any light. (Most people who use the term "blindness" mean complete blindness.)

## Physics Revision: Power of Lenses. The Human Eye Summary

There is a quantity used to determine how able is a lens to collect rays at points near it. This quantity is the reciprocal of focal length and is known as (optical) Power of a lens, P. The only condition is that focal length must be written in metres. Its formula therefore is:

P = 1/F (in metres)

Power of lens is measured in Dioptre [D].

When two lenses are in contact, the focal length of this system is calculated by the formula

1/F = 1/F1 + 1/F2

Therefore, the optical power of such a system of lenses, is

P = P1 + P2

When two lenses are not in contact, we must also consider the distance between them in order to calculate the optical power of this system. The formula of optical power in this case becomes

P = P1+P2-d × (P1 × P2)

where d is the distance (in metres) between the two lenses and P1 and P2 are the individual optical powers.

The eye has a spherical shape called eyeball, most of which has a white colour when healthy. The cornea is the clear front surface of the eye. It lies directly in front of the iris and pupil, and it allows light to enter the eye.

Attached to the pupil there is a converging lens, which refracts the light rays entering the eye, forming an inverted image on the retina, at the back of the eyeball.

The optic nerves attached to the retina carry these signals from the eye to the brain. The brain then converts the signals into upright image.

Ciliary muscle is a circular muscle that enables the lens to change shape for focusing.

The inside of the eyeball is filled with a watery liquid.

Light intensity entering the eye is controlled by the iris. If there is too much light, the iris enlarges, making the pupil smaller. In this way, the light intensity entering the eye decreases. This occurs during the day, in places exposed to sunlight. During the night the reverse process does occur, i.e. the iris shrinks, making the pupil larger. As a result, more light enters the eye and therefore, the light intensity increases.

The eye lens is flexible, i.e. it becomes thicker or thinner as needed. This is made possible by the action of ciliary nerves as stated earlier. When the eye lens changes thickness, its focal length also changes. Thus, the eye can make focusing adjustments for objects at various distances. This property of eye is known as eye accommodation.

Practically, a person with normal vision has an infinite distance of eye accommodation because he/she may see objects at very high distances without having any trouble. Therefore, the far point of normal people is at infinity. On the other hand, the nearest point in which the eye can accommodate effortlessly is known as the near point. For normal people the near point is 25 cm. This means the eye can be damaged when we try to see objects closer than 25 cm, because this causes tension in the eyes.

Many people have trouble in seeing clearly the objects around them. Their vision is blurred when it should not be so. This phenomenon occurs due to many factors, both inherited and acquired, and it is known as eye defect.

Basically, there are six types of eye defects:

1. Long-sightedness (Hyperopia). This occurs when light rays converge behind the retina. Converging lenses are used to correct this pathology.
2. Near-sightedness (Myopia). It occurs when light rays converge before the retina. Diverging lenses are used to correct this pathology.
3. Colour-blindness. This is another anomaly of the eye, in which the patient is not able to distinguish the colours of objects.
4. Astigmatism. This pathology is a result of an irregularly shaped cornea or eye-lens. If the eye-lens or cornea are not perfectly spherical, the eye can form a correct image only in some directions, but not in others. To correct this defect, cylindrical lenses are typically used.
5. Cataract. Cataract occurs when the vision becomes cloudy. Such a pathology can be corrected through a surgical intervention.
6. Blindness. It represents a pathology involving a lack of vision. It may also refer to a loss of vision that cannot be corrected with glasses or contact lenses. Partial blindness means the patient has very limited vision. Complete blindness means the patient cannot see anything and is not able to see any light.

## Physics Revision Questions for Power of Lenses. The Human Eye

1. An optician prescribes spectacles to a patient with a combination of a converging lens of focal length 40 cm and diverging lens of focal length 25 cm. What is the power of these spectacles?

1. 6.0 D
2. -6.0 D
3. 1.5 D
4. -1.5 D

2. A student wants to construct an optical system made by two identical converging lenses with focal lengths of 20 cm each. How far from each other must he place the lenses to obtain an optical power of -10 D for this system?

1. 0.8 m
2. 3.96 m
3. 3960 m
4. 247.5 m

3. The typical distance between the eye-lens and retina is 2.5 cm. A patient goes to an oculist who discovers that the light rays converge at a distance of 2.3 cm from the eye-lens, as shown in the figure. Which is correct for the lens the doctor prescribes the patient to correct his sight if the optical glasses must have 1 cm distance from the eye-lens?

1. Converging lens of focal length 16.25 cm
2. Diverging lens of focal length 6.15 m
3. Diverging lens of focal length 16.25 cm
4. Diverging lens of focal length 6.15 cm