Note on Real and Apparent Depth, Total Internal Reflection and Critical Angle

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Real Depth and Apparent Depth

An object placed in a denser medium, when viewed from rarer medium appears to be at a lesser depth than its real depth due to refraction of light.

Consider a point object O at the bottom of a beaker containing water. Suppose XY is the plane surface which separated air and water. A ray OA from O incident normally to the surface XY passes without bending along AD. Another ray OB is refracted away from normal along BC. When viewed from above, the rays will appear to come from point I which is the point of intersection of OD and BC produced backward. Thus, a virtual image of O is formed at I. Therefore, new depth the object is AL. The depth AO is the real depth of the object, and depth AI is called apparent depth which is as shown in the figure.

\begin{align*} \text {From Snell’s law} \\ _a\mu _w &= \frac {\sin r}{\sin i} \\ \text {where I is angle of incident and r is the angle of refraction. } \\ \text {In} \Delta ABO, \: \sin I &= \frac {AB}{OB} \\\text {In} \Delta ABI, \: \sin r &= \frac {AB}{IB} \\ \text {Then,} \\ _a\mu _w &= \frac {\sin i}{\sin r} \\ &= \frac {AB/IB}{AB/OB} \\ \text {If point B is very close to A, then we have } \: OB = OA \text {and} \: IB = IA. \text {So,} \\ _a\mu _w &= \frac {OA}{IA} = \frac {\text {real depth}} {\text {Apparent depth}} \dots (i) \end{align*}

Apparent shift

If the real depth, OA = t, then from equation (i), apparent depth \( = \frac {1}{_a\mu _w} \). The apparent shift of the object is given by

\begin{align*} d &= OI = OA-AI \\ &= \text {real depth} - \text {apparent depth} \\ &= \frac {1}{_a\mu _w} \\ \text {or,} &= t \left ( 1-\frac {1}{_a\mu _w} \right ) \\ \end{align*}

Total Internal Reflection and Critical Angle

When a ray of light passes from a denser medium to rarer medium, it bends away from the normal. Thus, an incident ray AO passes from water to air and bends away from normal in the direction OB. As the angle of incidence is increased, the angle of refraction also increases and becomes 90o at a particular angle of incidence. The angle of incidence in denser medium for which the angle of refraction in rarer medium is 90o, is called the critical angle C of the medium.

\begin{align*} \text {At this condition,} \\ _w\mu _a &= \frac {\sin i}{\sin r} \\ &= \frac {\sin c}{\sin 90^o} \\ &= \sin c \\ \text {or,} \: _a\mu _w &= \frac {1}{\sin c} \\ \therefore c &= \sin ^{-1} \left (\frac {1}{_a\mu _w}\right ) \\ \end{align*}

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If the angle of incidence in the denser medium is less than the critical angle, the light is refracted in the normal way. But is the angle is greater than the critical angle, the light is totally internally reflected in the same denser medium? This is known as total internal reflection. In the figure, AH represents a ray of light whose angle of independence (ÐAHN) is greater than critical angle c and HK is the reflected ray in the same medium which obeys the laws of reflection of light.

Conditions for Total Internal Reflection

  1. Objects must be in the denser medium.
  2. The angle of incidence should be greater than the critical angle.

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Some examples of total internal reflection:
  1. Mirage
    Mirage is an optical illusion produced on desert during hot summer days. The layer of air near the surface becomes very hot and the other layers above it gradually cooler. So the refractive index of different layers varies and this decreases from higher layers to lower one.
  2. Total reflecting prism
    Total reflecting prism is an isosceles right-angled prism made of crown glass. When a ray of light is incident on a face and reflected ray in another face makes the angle with normal greater than the critical angle, it will totally internally reflected. This prism binoculars and submarine periscopes use such types of prisms.
  3. Brilliancy of diamond
    Diamond shines because of multiple internal reflections occurs in it due to it’s cutting. The refractive index of diamond is 2.4 and hence, its critical angle is small, 24.60.
  4. Optical fibers
    Optical fibers are used to transmit light and communication signal through total internal reflection. It consists of a very fine, thin glass rod of diameter about 125 µm and has a central glass core surrounded by a glass coating and cladding of smaller refracting index than the core. When the small angle of the incident at one end passes through the fiber, it undergoes total multiple reflections at the boundary of core and cladding and finally, it comes out of the fiber at another end without much loss of intensity.

Applications

  1. Optical fibers are used in telecommunications.
  2. Used to transmit light to places where it is difficult to reach.
  3. The fibers are used to view the internal organs of human body.
  4. Used to carry a computer, telephone and television signals, in the form of laser pulses.

 

  • $$From\; Snell’s\; law\;  _a\mu _w  = \frac {\sin r}{\sin i} $$ 
  • An object placed in a denser medium, when viewed from rarer medium appears to be at a lesser depth than its real depth due to refraction of light.
  • Conditions for Total Internal Reflection

    1. Objects must be in the denser medium.
    2. The angle of incidence should be greater than the critical angle.
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