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Geometric optics: what it studies, laws, applications, exercises

We explain Geometric optics and what it studies, laws, applications and exercises. The geometrical optics is the branch of physics that focuses on studying how light spreads and is reflected when it passes from one medium to another, regardless of the effects of diffraction. Geometric optics

In this way, light is represented geometrically by rays, imaginary lines perpendicular to the light wave fronts.

Geometric optics: what it studies, laws, applications, exercises
Geometric optics: what it studies, laws, applications, exercises

Rays of light emerge from light sources such as the Sun , a flame or a light bulb, spreading in all directions. Surfaces partly reflect these rays of light and that is why we can see them, thanks to the fact that the eyes contain elements that are sensitive to light.

Thanks to the ray treatment, geometric optics does not take so much into account the wave aspects of light, but rather explains how images are formed in the eye, mirrors and projectors, where they do it and how they appear.

The fundamental principles of geometric optics are the reflection and refraction of light. The light rays strike at certain angles on the surfaces with which they are, and thanks to this a simple geometry helps to keep track of their trajectory in each medium.

This explains everyday things like looking at our image in the bathroom mirror, seeing a teaspoon that seems to bend inside a glass full of water, or improving vision with proper glasses.

We need light to relate to the environment, that is why, since always, its behavior has amazed observers, who wondered about its nature.

What does geometric optics study? (Object of study)

Geometric optics studies the propagation of light in a vacuum and in various media, without explaining what its true nature consists of. For this it makes use of the ray model and simple geometry.

A ray is the path that light follows in a certain transparent medium, which is an excellent approximation as long as the wavelength is small compared to the size of objects.

This is true in most of the everyday cases, such as those mentioned at the beginning.

There are two fundamental premises of geometric optics:

-The light propagates in a rectilinear way.

-While it propagates through various means, light does so following empirical laws, that is, obtained from experimentation.

Basic concepts in geometric optics

Refractive index

The speed of light in a material medium is different from that of a vacuum. There we know that it is 300,000 km / s, but in the air it is just a little lower, and even more so in water or glass.

The refractive index is a dimensionless quantity, which is defined as the quotient between the speed with which light travels in a vacuum o and the speed c   in said medium:

n = c o / c

Optical path

Geometric optics: Optical path
Optical path

It is the product between the distance traveled by light to pass from one point to another, and the refractive index of the medium:

L = s. n

Where L is the optical path, s is the distance between the two points and n represents the refractive index, assumed constant.

By means of the optical path, light rays moving in different media are compared.

Angle of incidence

Geometric optics: Here the angle of incidence is called θ 1.
Here the angle of incidence is called θ 1.

It is the angle that the light ray forms with the normal line to a surface that separates two media.

Laws of Geometric Optics

Fermat’s principle

Geometric optics: Fermat's principle in the case of the refraction of light on a plane surface between air and water. Object point A in air and observation point B in water. The point of refraction P is the one that minimizes the time it takes for light to travel the APB path.
Fermat’s principle in the case of the refraction of light on a plane surface between air and water. Object point A in air and observation point B in water. The point of refraction P is the one that minimizes the time it takes for light to travel the APB path.

The French mathematician Pierre de Fermat (1601-1665) noted that:

When a ray of light travels between two points, it follows the path in which it takes the least amount of time.

And since light moves with constant speed, its path must be rectilinear.

In other words, Fermat’s principle states that the path of the light beam is such that the optical path between two points is minimal.

Law of reflection

Law of reflection
Law of reflection

When striking the surface that separates two different media, a part of the incident ray – or all of it – is reflected back and does so with the same measured angle with respect to the normal to the surface with which it struck.

An example of the law of reflection.
An example of the law of reflection.

In other words, the angle of incidence equals the angle of reflection:

 θ i = θ i ‘

Snell’s Law

Snell's law.
Snell’s law.

The Dutch mathematician Willebrord Snell (1580-1626) carefully observed the behavior of light as it passes from air to water and glass.

He saw that when a ray of light falls on the surface that separates two media, forming a certain angle with it, one part of the ray is reflected back towards the first medium and the other continues its way through the second.

Thus he deduced the following relationship between both media:

1 ⋅ sin θ 1 = n 2 ⋅  sin θ 2

Where n 1 and n 2 are the respective refractive indices , while θ 1 and  θ 2  are the angles of incidence and refraction, measured with respect to the normal to the surface, according to the figure above.

Applications Geometric optics

Mirrors and lenses

Lenses are devices based on geometric optics that are used, among other things, to improve vision.
Lenses are devices based on geometric optics that are used, among other things, to improve vision.

Mirrors are highly polished surfaces that reflect light from objects, allowing images to be formed. Flat mirrors, such as those in the bathroom or those you carry in your purse, are common.

A lens consists of an optical device with two very close refractive surfaces. When a beam of parallel rays passes through a converging lens, they converge at a point, forming an image. When it comes to a diverging lens, the opposite happens: the beam’s rays diverge on the dot.

Lenses are frequently used to correct refractive errors in the eye, as well as in various optical magnifying instruments.

Optical instruments

There are optical instruments that allow images to be magnified, for example microscopes, magnifying glasses and telescopes. There are also those for looking above eye level, like periscopes.

Photographic cameras are used to capture and preserve images, which contain a lens system and a recording element to save the image formed.

Fiber optic

It is a long, thin and transparent material based on silica or plastic, which is used for data transmission. It takes advantage of the property of total reflection: when the light reaches the medium at a certain angle, no refraction occurs, therefore the ray can travel long distances, bouncing inside the filament.

Exercise resolved

Objects at the bottom of a pool or pond appear to be closer than they actually are, which is due to refraction. At what apparent depth does an observer see a coin that is at the bottom of a 4 m deep pool? Geometric optics

Suppose that the ray emerging from the coin reaches the observer’s eye at an angle of 40º with respect to the normal.

A coin at the bottom of the pool looks closer when viewed from above.
A coin at the bottom of the pool looks closer when viewed from above.

Fact: the refractive index of water is 1.33, that of air is 1.

Solution Geometric optics

The apparent depth of the coin is s’ and the depth of the pool is s = 4 m. The coin is at point Q and the observer sees it at point Q ‘. The depth of this point is:

s´ = s – Q´Q

From Snell’s law:

b ⋅ sin 40º = n a ⋅ sin θ r

sin θ r = (n b ⋅ sin 40º) ÷ n a  = sin 40º /1.33 = 0.4833

θ r = arcsen (0.4833) = 28.9º

Knowing this angle, we calculate the distance d = OV from the right triangle, whose acute angle is  θ r :

tan 28.9º = OV / 4 m

OV = 4m × tan 28.9º = 2.154 m

On the other hand:

tan 50º = OQ´ / OV

Therefore:

OQ´ = OV × tan 50º = 2.154 m × tan 50º = 2.57 m.

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