Recently I working on a optics Question. Then , An Confusion is created in my mind , So please Help me to get rid out from that !

Problem: If Light ray is incident at an angle \(\alpha \) on an glass slab which has variable refractive index which varies as : \(\mu (y)\quad =\quad { \mu }_{ 0 }\quad -\quad Ky\). where K : is constant and refractive index of outside medium is \({ \mu }_{ 0 }\) Then Find maximum Hight at which Light ray goes up ?

Solution: Using Snell's Law at intial state and Final State : \(\\ \quad { \mu }_{ 0 }\sin { \alpha } \quad =\quad ({ \mu }_{ 0 }\quad -\quad K{ h }_{ max })\sin { 90 } \\ \\ { h }_{ max }\quad =\quad \cfrac { { \mu }_{ 0 }(1-\sin { \alpha } ) }{ K } \).

**Doubt** :

1)- **What is the critical angle in this situation ? Can we say it as 90 degree ?**

2)- **How can we use snell's Law at Particular an intermediate Point ? Is always
i = r ?**

Expln-1 ) -Because if we use **Snell's Law** at the interface i.e

\(\mu \sin { { \theta }_{ C } } =({ \mu }_{ 0 }\quad -\quad K{ h }_{ max })\sin { 90 } \quad \quad \).

also \(\mu \quad =\quad ({ \mu }_{ 0 }\quad -\quad K{ h }_{ max })\). because at that point hight is almost same !

So this gives \({ \theta }_{ C }\quad =\quad 90\).

Expln-2) - SInce at particular point refractive index of medium is same at just below and above of the interface. So By snell's Law incident angle (i) = refracted angle (r)

**I know this wrong But Please Tell me Where I'am wrong ??**

Thanks a lot !!

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## Comments

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TopNewestTo solve this problem we model the glass slab as a collection of infinitely many extremely thin glass slabs stacked together. We also assume that since the thickness of each of these slabs is so small their refractive index does not vary much and light follows a straight path inside them. Applying Snell's Law at an interface:

\(\mu(y)\sin i=\mu(y+\text{d}y)\sin r\)

Thus from the equation above it is clear that \(r\neq i\) but the value of \(r\rightarrow i\).

By geometry this value of \(r\) is same as the angle of incidence at the next interface. As the angle of incidence has changed by a small amount let us call it \(i+\text{d}i\).

\(\therefore (\mu_{0}-ky)\sin i=(\mu_{0}-k(y+\text{d}y))\sin (i+\text{d}i)\)

Simplifying we obtain:

\(\dfrac{\text{d}i}{\text{d}y}=\dfrac{k}{\mu_{0}-ky}\tan i\)

Solving this differential equation we obtain the answer

\(h_{max}=\dfrac{\mu_{0}(1-\sin \alpha)}{k}\)

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@Deepanshu Gupta - you see , initially you have assumed that light is not incident at the critical angle of the base (y=0) Now as you go above the refractive index falls, but it falls extremely slowly, in any small interface,, you can see that if critical angle is reached it would mean

\(\frac { { \mu }_{ 2 } }{ { \mu }_{ 1 } } =\frac { sin(i) }{ sin(r) } \\ \\ \frac { \mu -kd(y) }{ \mu } =1-\frac { kd(y) }{ \mu } =\frac { sin({ i }_{ c }) }{ sin(90) } =sin({ i }_{ c })\)

clearly as dy--> 0

\(\frac { { \mu }_{ 2 } }{ { \mu }_{ 1 } } =\frac { \mu -kd(y) }{ \mu } =1-\frac { kd(y) }{ \mu } -->\quad 1\\ \)

so effective critical angle at any interface is 90 degrees (and it is exactly 90 degrees if the transition is absolutely smooth theoretically)

thus no internal reflection will occur till i becomes 90 degrees which is what you have used to get the maximum height, That is i gradually rises till critical angle 90 degrees

so no snells law has not been violated,, just that due to smooth gradation,,, internal reflection only occurs when ray has become paralell to the ground (which is the max height case)

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Yeah ! Now I completely Got it ! Thanks a lot @Mvs Saketh !!

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Hey depanshu can you plz guide me about some good books to excel in maths

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But Sometimes I refer "**Arihant New pattern IIT JEE Book" , But I'am quite uncertain about maths stuff to give you any advice , Since I didn't solve other book's ! So I'am unable to help You as much !

But I Specially Recommend You To Buy "Playing with graphs" By Amit agarwal , it is really very good book's to read !

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Kartik's solution is best and easy, in these type of problems light instead of travelling in straight line it travels in curve I can give that curve with another method

Suppose light is incident at origin (from 3rd quadrand) and glass is in 1st quadrant with y-axis as normal

\(\mu_0 \sin \alpha = \mu sin (r)\)

\(\displaystyle sin (r) = \frac{\mu_0 \sin \alpha}{\mu}\)

We can say that

\(\frac{dy}{dx} = cot (r)\)(slope of curve)

cot(r) can be easily found(given sin(r)) to obtain trajectory

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where i can found a book with problems in mechanics

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I think we cannot determine the max height!! See the explanation

I agree with karthik and saketh that max height is reached when r=90 but after that ,total internal reflection occurs which makes the light go further upward and again the same procedure continues .So there is no max height

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no bro after total reflection light ray will come back downward not upward ! and finally it will come with same angle of incidence as that of initial . it's graph is symmetric about about the critical point !

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