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Sign convention

The rules of refraction of the light through curved surfaces are different from the rules of the refraction of the light through a plain surface. Optic Centre of the curved surface is taken as reference to measure all the distances. The incident light is always considered from left to right in the given diagram.

If the object distance and the image distance are measured along the direction of incident light, they are treated like positive values. If the distance of the object or image is measured against the direction of the incident light, it shall be treated as negative. This is called sign convention. Throughout the Ray optics,we are going to follow the same sign convention.

Law of refraction of the light at the curved surfaces is represented as shown in the attached diagram.

A lens is a piece of a transparent material which has two refracting surfaces such that at least one of them is curved. The refractive index of the lens material shall be different from the surroundings refractive index.

There are different types of lenses like double convex lens, Plano convex lens, double concave lens and Plano concave lens. The double convex lens consists of two convex surfaces whereas the Plano convex lens consists of only one convex surface. The double concave lens consists of two concave surfaces and the plano concave lens consists of only one concave surface.

While solving all the problems that are relevant to lens, we assume that light is falling from left to the right in the given diagram or in a given situation. This is a default way of consideration and it is called sign convention. In the lens formula or in the lens makers formula, we shall write the known values with the sign convention. The known values have to be written as it is that are there in the formula without applying the sign convention. The answer will decide the corresponding sign convention.

Any curved portion of the lens is drawn from a sphere. The radius of the corresponding sphere is called radius of curvature. Radius of curvature is mathematically double to the value of the focal length. The light rays after passing through the lens will converge at a particular point and that particular point is called principal focus. The distance between optics Centre and the principal focus is called focal length. The radius of curvature that is measured along the direction of the incident light is treated as positive and vice versa.

To know the relation between object distance, image distance and the focal length of a lens we are having a formula called Lens formula.

To make a lens with a particular focal length, we can use a formula called lens makers formula. The focal length of a lens depends on the refractive index of the material of the lens, refractive index of the surroundings of the medium and the radius of curvature of the curved surfaces. For a plain surface the radius of curvature is infinite.

We can write the lens makers formula for the concave lens as shown.

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When a light ray moves from denser medium to rarer medium, it moves away from the normal. This is just because the different mediums are having different refractive index. When there is a change of media, the wavelength and the velocity of the light changes. As a consequence its path is also modified. With respect to the increase of angle of incidence, angle of refraction also increases.

At a particular angle of incidence, the refracted light ray grazes the boundary that is separating the two media. This particular angle of incidence is called critical angle. If the angle of incidence is more than the critical angle, the light ray reflects back into the same medium. This phenomenon is called total internal reflection.

Problem and solution

A ray of light travelling in a transparent medium of known refractive index , falls on the surface separating the medium at an angle of incidence of 45°. Find the value of the refractive index at which the light ray experience total internal reflection?

We know that for the total internal reflection, the angle of incidence shall be more than that of critical angle. Taking the very basic concept of this into consideration,we can solve the problem as shown below.

Problem and solution

The speed of light into different media is given. A ray of light enters from medium 1 to medium 2 it and angle of incidence i. If the light ray suffers total internal reflection, what is the value of angle of incidence?

As it is explained earlier, whenever there is a change of medium there will be a change of wavelength as well as the velocity of light. Frequency is the characteristic property of the source and it remains constant even when there is a change of medium.

Here in this problem being the media are different the refractive index is also automatically different. For the total internal reflection to happen, the light ray shall always moves from denser medium to rarer medium. Velocity of the light is more in the rarer medium and less in the denser medium. Velocity of light is maximum in vacuum because that is the rarest medium. The refractive index of vacuum is treated as one.

Basing on the definition of the total internal reflection we can write refractive index of the denser medium to rarer medium as shown. Being frequency is constant, the ratio of refractive index of the two media is inversely proportional to their respective velocities of light.

The solution to the problem is as shown below.

Problem and solution

If a ray of light in the denser medium strikes a rarer medium and angle of incidence, the angle of reflection and angle of refraction are given. If the reflected and the reflected light rays are at right angles to each other, the critical angle for the given pair of the media is how much?

As the light ray is moving from denser medium to rarer medium angle of refraction is more than that of angle of incidence. After striking the boundary some portion of the light reflects back into the same medium and some another portion of the light refracts to the other medium.

It is given in the problem that the reflected and the reflected light rays are at right angles to each other. By applying the basic mathematics and the definition of the critical angle with can solve the problem as shown below.

Problem and solution

A beam of light consists of red, green and blue colors. This light incidents on the right angled prism as shown. The refractive indices of the materials of the prism for the different colors are given. Find the color which will be separated from the other colors?

As each color has different refractive index, each color will have different critical angle when it is passing through the same glass prism. For the color which has an angle of incidence is more than the critical angle of the prism, there will be total internal reflection. It is clear basing on the deformation of the critical angle that for the total internal reflection to happen the refractive index of the color shall be greater than 1.414.

Basing on the values of the refractive indices it is clear that green and blue colors experience total internal reflection and the red refracts into the other medium.