# Refraction at plane surface

Critical angle:

The angle of incidence in denser medium for which the angle of refraction in rarer is 90 degree, is called the critical angle of the medium

Light pipe:

Light pipes are physical structures used for transporting or distributing natural or artificial light for the purpose of illumination, and are examples of optical waveguides.

Uses:

1. Study of internal organ

2. Use in laboratory purposes.

3. They are used in special mirror.

Optically denser medium;

The materials listed at the bottom of the table are those through which light travels slowest; these are the most optically dense materials. So as the index of refraction value increases, the optical density increases, and the speed of light in that material decreases.

Angle of refraction and refractive index:

For the given media, the ratio of the sine of the angle of incidence to the sin e of the angle of refraction is constant, called refractive index and is denoted by µ.

aµb = $\frac{{\sin {\rm{i}}}}{{\sin {\rm{r}}}}$

Where aµb is the refractive index of medium b is with respect to medium a. when light travels through vacuum to any other medium, the refractive index is called absolute refractive index.

A practical application of total internal reflection:

The cut of the diamond favors total internal reflection.  Most rays entering the top of the diamond will internally reflect until they reach the top face of the diamond where they exit.  A fiber optic is a glass "hair" which is so thin that once light enters one end, it can never strike the inside walls at less than the critical angle.   The light undergoes total internal reflection each time it strikes the wall.   Only when it reaches the other end is it allowed to exit the fiber.

1. Fiber optics can carry much more information in a much smaller cable.

2. No interference from electromagnet fields result in "clearer" connections.

3. No electrical resistance.

4. No hazard of electrocution if cable breaks.

Mirage;

The mirageis an optical illusion caused by atmospheric conditions, especially the appearance of a sheet of water in a desert or on a hot road caused by the refraction of light from the sky by heated air.

A relation between refractive index and critical angle of the medium:

The angle of incidence in denser medium for which the angle of refraction in rarer is 90 degree, is called the critical angle of the medium

If the angle of incidence is greater than or equal to the critical angle, the ray of light is reflected following the laws of reflection. This phenomenon of light is known as total internal reflection.

Suppose that a ray of light originates from an underwater source and travels upwards. At the surface part of it will be refracted into air and part will be reflected back into the water. The angle of reflection equals the angle of incidence, as shown, while the angle of refraction must be consistent with Snell's law:

n(w)sin(i) = n(a)sin(r)……………..1

Where n(w) and n(a) are the indices of refraction of water and air respectively. With n(w)=1.33 and n(a)=1.00 we have

sin(r) = (1.33)sin(i)……………2

At normal incidence (i=0 degrees) the transmitted ray is undeviated and the reflected ray travels directly back to the source. Increasing the angle of incidence, i.e. making the ray meet the water surface more obliquely, causes the refracted ray to bend more towards the surface. This continues until the angle of refraction reaches its maximum value of 90 degrees, corresponding to the critical angle of incidence 'c' given by the equation (1.33)sin(c)=1, or c=48.8 degrees. For angles of incidence greater than the critical angle there is total internal reflection.

total internal refraction

Derive refractive indices for different media in contact:

For the given media, the ratio of the sine of the angle of incidence to the sin e of the angle of refraction is constant, called refractive index and is denoted by µ.

aµb = $\frac{{\sin {\rm{i}}}}{{\sin {\rm{r}}}}$

Where aµb is the refractive index of medium b is with respect to medium a. when light travels through vacuum to any other medium, the refractive index is called absolute refractive index. The ratio of the velocity of light is vacuum to the velocity of light in that medium is defined as the refractive index of medium.

Absolute refractive index, ${\rm{\mu }} = \frac{{{\rm{velocity\: of\: light\: in\: vaccum}}}}{{{\rm{velocity\: of\: light\: in\: medium}}}}$

$= \frac{{\rm{c}}}{{\rm{v}}}$

It depends upon density of the medium.

Suppose that a ray of light originates from an underwater source and travels upwards. At the surface part of it will be refracted into air and part will be reflected back into the water. The angle of reflection equals the angle of incidence, as shown, while the angle of refraction must be consistent with Snell's law:

n(w)sin(i) = n(a)sin(r)……………..1

Where n(w) and n(a) are the indices of refraction of water and air respectively. With n(w)=1.33 and n(a)=1.00 we have

sin(r) = (1.33)sin(i)……………2

At normal incidence (i=0 degrees) the transmitted ray is undeviated and the reflected ray travels directly back to the source. Increasing the angle of incidence, i.e. making the ray meet the water surface more obliquely, causes the refracted ray to bend more towards the surface. This continues until the angle of refraction reaches its maximum value of 90 degrees, corresponding to the critical angle of incidence 'c' given by the equation (1.33)sin(c)=1, or c=48.8 degrees. For angles of incidence greater than the critical angle there is total internal reflection.

total internal refraction

Lateral shift in terms of thickness of a slab and the angle of incidence of light:

The shift of a light ray from its original path when passing through a slab is known as lateral shift. It is denoted by d.

lateral shift

From ΔABQ, we have

$\sin \left( {{\rm{i}} - {\rm{r}}} \right) = \frac{{{\rm{BQ}}}}{{{\rm{AB}}}}$

Or, BQ = AB sin (I – r)……1

Again, from ΔABN2, we have,

$\cos {\rm{r}} = \frac{{{\rm{A}}{{\rm{N}}_2}}}{{{\rm{AB}}}} = \frac{{\rm{t}}}{{{\rm{AB}}}}$

Or, ${\rm{AB}} = \frac{{\rm{t}}}{{{\rm{cos\: r}}}}$

So, equation 1 becomes

$\cos {\rm{r}} = \frac{{{\rm{A}}{{\rm{N}}_2}}}{{{\rm{AB}}}} = \frac{{\rm{t}}}{{{\rm{AB}}}}$

${\rm{BQ}} = \frac{{{\rm{t}}\sin \left( {{\rm{i}} - {\rm{r}}} \right)}}{{\cos {\rm{r}}}}$

Hence when I = 90

`
${\rm{BQ}} = {\rm{d}} = \frac{{{\rm{t}}\sin \left( {90 - {\rm{r}}} \right)}}{{\cos {\rm{r}}}}$

$= \frac{{{\rm{t\: cos\: r}}}}{{{\rm{cos\: r}}}}$

Or, d = t

It means that when light is incident at 90 on the surface of glass slab, lateral shift produced by it is equal to the thickness of the slab.

Real and apparent depth:

µ = $\frac{{{\rm{Real\: depth}}}}{{{\rm{Apparent\: depth}}}}$.

The actual depth of the object is called the real depth. And the depth seen by the observer is called apparent depth.

Figure:  apparent depth

From Snell’s law

aµw = $\frac{{\sin {\rm{r}}}}{{\sin {\rm{i}}}}$

Where i is angle of incident and r is angle of refraction.

In Δ ABO,

$\sin {\rm{i}} = \frac{{{\rm{AB}}}}{{{\rm{OB}}}}$

And in ΔABI,

$\sin {\rm{r}} = \frac{{{\rm{AB}}}}{{{\rm{IB}}}}$

Then,

aµw = $\frac{{\sin {\rm{r}}}}{{\sin {\rm{i}}}}$

aµw =  $\frac{{\frac{{\frac{{{\rm{AB}}}}{{{\rm{IB}}}}}}{{{\rm{AB}}}}}}{{{\rm{OB}}}}$

$= \frac{{{\rm{OB}}}}{{{\rm{IB}}}}$

If point B is very close to A, Then we have OB = OA and IB = IA. So,

µ = $\frac{{{\rm{Real\: depth}}}}{{{\rm{Apparent\: depth}}}}$.

Terms total internal reflection and critical angle:

If the angle of incidence in the denser medium is less then critical angle, the light is refracted in the normal way. But if the angle of incidence is greater than critical angle, the light is totally internally reflected in same denser medium. This is known as total internal reflection.

The angle of incidence in denser medium for which the angle of refraction in rarer is 90 degree, is called the critical angle of the medium

Applications are as follows;

1. Optical fibers are used in telecommunications.

2. Used to transmit light to places where it is difficult to reach.

3. The fibers are flexible and can be used to view the internal organs of the human body without surgery.

4. Used to carry computer, telephone and television signal in the form of laser pulses.

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