The critical angle of light going from medium A into medium B is \$\theta \$ . If the speed of light in medium A is \$v\$, then the speed of light in medium B is :

For total internal reflection \$n_{A} sin \theta =n_{B}\$when \$n_{A}=\$ refractive index of medium A\$n_{B}=\$ refractive index of medium B\$\dfrac{n_{A}}{n_{B}}=\dfrac{1}{sin \theta}\$by snell's law \$\dfrac{n_{A}}{n_{B}}=\dfrac{v_{B}\left ( velocity \ in \ medium \ B \right )}{v_{A}(velocity \ in \ medium \ A)}\$\$\dfrac{v_{B}}{v_{A}}=\dfrac{1}{sin \theta}\$\$v_{B}=\dfrac{v_{A}}{sin \theta}\$\$=\dfrac{v}{sin \theta}\$

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Critical Angle - Formula

Q: A silicon optical fibre with a core diameter large enough has a core refractive index of 1.50 and a cladding refractive index 1.47. Determine(i) the critical angle at the core cladding interface,(ii) the numerical aperture for the fibre(iii) the acceptance angle in air for the fibre.

Here,\${ \mu }_{ 1 }=1.50;{ \mu }_{ 2 }=1.47;\$\${ \mu }_{ 0 }=1\$(i) Critical angle \${ \theta }_{ c }\$ at the core-cladding interface is given by \${ \theta }_{ c }={ sin }^{ -1 }\dfrac { 1.47 }{ 1.50 } =78.{ 5 }^{ 0 }\$(ii) Numerical aperture,\$NA={ \left( { { \mu }_{ 1 } }^{ 2 }-{ { \mu }_{ 2 } }^{ 2 } \right) }^{ \frac { 1 }{ 2 } }\$ \$={ \left[ { \left( 1.50 \right) }^{ 2 }-{ \left( 1.47 \right) }^{ 2 } \right] }^{ \frac { 1 }{ 2 } }={ \left( 2.25-2.16 \right) }^{ 1/2 }=0.30\$(iii) Acceptance angle \${ \theta }_{ a }={ sin }^{ -1 }\left( NA \right) \$\$={ sin }^{ -1 }\left( 0.30 \right) ={ 17.4 }^{ 0 }\$

Q: A fish looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is 4/3 and the fish is 12cm below the surface, the radius of this circle (in cm) is:

\$\displaystyle \sin i_{c}=\frac{1}{\mu}=\frac{r}{\sqrt{r^{2}+h^{2}}}\$Using \$h=12\$ cm, \$\mu=4/3\$We get \$\displaystyle r=\frac{36}{\sqrt{7}}\$ cm.

Q: The refractive index of the material of a prism is \$\sqrt{2}\$ and the angle of the prism is \$30^o\$. One of the two refracting surface of the prism is made a mirror inwards, by silver coating. A beam of monochromatic light entering the prism from the other face will retrace its path (after reflection from the silvered surface) if its angle of incidence on the prism is?

Using \$ A = r_{1} + r_{2}\$\$ 30 = r_{1} + 0\$so \$r_{1} = 30\$Using Snell's Law, \$ 1 \times sin i = \mu \times sin r\$\$ 1 \times sin i = \sqrt{2} \times sin 30 = \sqrt{2} 3/2\$\$sin i = 1/ \sqrt{2}\$\$ i = 45^{o}\$option 4 is correct

Q: Mention three applications of total internal reflection of light.

Total internal reflection can be applied in the following:1. Telecommunication systems2. Automotive rain sensors and windscreen wipers3. Optical fingerprinting devices

Q: What are the necessary conditions for total internal reflection to take place?

For total internal reflection to take place (i) light must travel from a denser medium to a rarer medium and (ii) the angle of incidence inside the denser medium must be greater than the critical angle. i.e., \$\angle i > \angle c\$.

Q: A ray of light travelling in a transparent medium of refractive index \$\mu\$, falls on a surface separating the medium from air at an angle of incidence of \$45^o\$ . For which of the following value of \$\mu\$ the ray can undergo total internal reflection ?

For total internal reflection, sin i > sin c where, i = angle of incidence, C = critical angle But, \$sin \, C \, = \, \dfrac{1}{\mu} \,\, \therefore \, sin \, i \, = \, \dfrac{1}{\mu} \,\, or \,\, \mu \, = \, \dfrac{1}{sin \, i}\$\$\therefore \, \mu \, > \, \dfrac{1}{sin \, 45} \,\, or \,\, \mu \, > \, \sqrt 2 \,\, (i \, = \, 45 \, Given)\$Hence, option (c) is correct.

Q: A ray of light is incident at the glass-water interface at an angle \$i\$. It merges finally parallel to the surface of the water. Then, the value of \$\mu_{g}\$ would be(where \$\mu_{g}\$ is the refractive index of glass with respect to water)

Applying Snell's law at glass-water surface,\$_{g}^{w}\mu = \dfrac {\sin r}{\sin i} = \dfrac {_{g}\mu}{_{w}\mu} ... (i)\$Applying Snell's law at water-air surface,\$_{w}^{a}\mu = \dfrac {\sin 90^{\circ}}{\sin r} = \dfrac {_{w}\mu}{_{a}\mu} ... (ii)\$Multiplying (i) and (ii),\$\dfrac {_{g}\mu}{_{a}\mu} = \dfrac {1}{\sin i}\ \Rightarrow _{g}\mu = \dfrac {1}{\sin i}\$.

Q: A transparent solid cylindrical rod has a refractive index of \$\displaystyle \dfrac{2}{\sqrt{3}}\$. It is surrounded by air. A light ray is incident at the mid-point of one end of the rod as shown in the figure. The incident angle \$\theta\$ for which the light ray grazes along the wall of the rod is -

\$ \displaystyle I\times \sin90 = \frac{2}{\sqrt{3}}\sin(90-\alpha)\$\$\Rightarrow \displaystyle \cos\alpha=\frac{\sqrt{3}}{2}\$So, \$\displaystyle \sin\alpha=\sqrt{1-\frac{3}{4}}=\frac{1}{2}\$Now, \$ 1\displaystyle \times\sin\theta=\frac{2}{\sqrt{3}}\sin\alpha=\displaystyle \frac{2}{\sqrt{3}}\times\frac{1}{2}=\frac{1}{\sqrt{3}}\$\$\Rightarrow \theta =\displaystyle \sin^{-1}\frac{1}{\sqrt{3}}\$

Physics

result

Factors affecting critical angle

Critical angle is affected by the refractive index of second medium with respect to the first medium. This in turn depends on:

Wavelength: Critical angle increases with increase in wavelength (least for violet).
Temperature: Critical angle increases with increase in temperature.

formula

Relation between refractive index and critical angle

Let us consider medium 1

-

Incident medium and 2

-

Refractive medium, Then value of critical angle can be derived by Snell's law.

n_{1} s i n θ_{i} = n_{2} s i n θ_{r}

n_{1} s i n θ_{c r i t} = n_{2} s i n 9 0^{\circ}

s i n θ_{c r i t} = \frac{n _{2}}{n _{1}}

s i n θ_{c r i t} = \frac{1}{n _{12}}

$n_{12}$ : Refractive index of denser medium 1 with respect to rarer medium 2.

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Total Internal Reflection and Critical angle

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Important Questions

A silicon optical fibre with a core diameter large enough has a core refractive index of 1.50 and a cladding refractive index 1.47. Determine
(i) the critical angle at the core cladding interface,
(ii) the numerical aperture for the fibre
(iii) the acceptance angle in air for the fibre.

Medium

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A fish looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is 4/3 and the fish is 12cm below the surface, the radius of this circle (in cm) is:

Medium

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The refractive index of the material of a prism is $2$ and the angle of the prism is $3 0^{o}$ . One of the two refracting surface of the prism is made a mirror inwards, by silver coating. A beam of monochromatic light entering the prism from the other face will retrace its path (after reflection from the silvered surface) if its angle of incidence on the prism is?

Medium

NEET

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Mention three applications of total internal reflection of light.

Easy

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What are the necessary conditions for total internal reflection to take place?

Medium

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A ray of light travelling in a transparent medium of refractive index $μ$ , falls on a surface separating the medium from air at an angle of incidence of $4 5^{o}$ . For which of the following value of $μ$ the ray can undergo total internal reflection ?

Medium

View solution

A ray of light is incident at the glass-water interface at an angle $i$ . It merges finally parallel to the surface of the water. Then, the value of $μ_{g}$ would be(where $μ_{g}$ is the refractive index of glass with respect to water)

Medium

JEE Advanced

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A transparent solid cylindrical rod has a refractive index of $\frac{2}{3}$ . It is surrounded by air. A light ray is incident at the mid-point of one end of the rod as shown in the figure. The incident angle $θ$ for which the light ray grazes along the wall of the rod is -

Hard

JEE Mains

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Define critical angle. Explain total internal reflection using a neat diagram.

Medium

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The critical angle of light going from medium A into medium B is $θ$ . If the speed of light in medium A is $v$ , then the speed of light in medium B is :

Medium

View solution

result

Factors affecting critical angle

formula

Relation between refractive index and critical angle

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Total Internal Reflection and Critical Angle

A silicon optical fibre with a core diameter large enough has a core refractive index of 1.50 and a cladding refractive index 1.47. Determine (i) the critical angle at the core cladding interface, (ii) the numerical aperture for the fibre (iii) the acceptance angle in air for the fibre.

A fish looking up through the water sees the outside world contained in a circular horizon. If the refractive index of water is 4/3 and the fish is 12cm below the surface, the radius of this circle (in cm) is:

Mention three applications of total internal reflection of light.

What are the necessary conditions for total internal reflection to take place?

A ray of light travelling in a transparent medium of refractive index μ, falls on a surface separating the medium from air at an angle of incidence of 45o . For which of the following value of μ the ray can undergo total internal reflection ?

A ray of light is incident at the glass-water interface at an angle i. It merges finally parallel to the surface of the water. Then, the value of μg​ would be(where μg​ is the refractive index of glass with respect to water)

A transparent solid cylindrical rod has a refractive index of 3​2​. It is surrounded by air. A light ray is incident at the mid-point of one end of the rod as shown in the figure. The incident angle θ for which the light ray grazes along the wall of the rod is -

Define critical angle. Explain total internal reflection using a neat diagram.

The critical angle of light going from medium A into medium B is θ . If the speed of light in medium A is v, then the speed of light in medium B is :

A silicon optical fibre with a core diameter large enough has a core refractive index of 1.50 and a cladding refractive index 1.47. Determine
(i) the critical angle at the core cladding interface,
(ii) the numerical aperture for the fibre
(iii) the acceptance angle in air for the fibre.

A ray of light travelling in a transparent medium of refractive index $μ$ , falls on a surface separating the medium from air at an angle of incidence of $4 5^{o}$ . For which of the following value of $μ$ the ray can undergo total internal reflection ?

A ray of light is incident at the glass-water interface at an angle $i$ . It merges finally parallel to the surface of the water. Then, the value of $μ_{g}$ would be(where $μ_{g}$ is the refractive index of glass with respect to water)

A transparent solid cylindrical rod has a refractive index of $\frac{2}{3}$ . It is surrounded by air. A light ray is incident at the mid-point of one end of the rod as shown in the figure. The incident angle $θ$ for which the light ray grazes along the wall of the rod is -

The critical angle of light going from medium A into medium B is $θ$ . If the speed of light in medium A is $v$ , then the speed of light in medium B is :