Apparent and Actual Depth__ Further more on Refraction of Light

 27th March 2013 , Wednesday

The Sun was bright and the day was blithesome. My Friends and I were splashing coloured water and throwing rainbow coloured powder everywhere , celebrating the radiant festival of Holi. After drowning eachother and pranking most of the adults by throwing coloured powders at them unexpectedly. We decided to spend the rest of the day in the pool, The bright sky, hot weather and fun in the air, made it perfect. We grabbed our towels and excitedly ran towards the pool, all the while getthing chided by the housekeeping aunty for making the once pristine corridor, into a form of abstract art drawn using our feet and the unsettled powder falling of our hair. As soon as we reached everyone jumped into the pool, doing cannonballs and splashing water all over. I quietly went and sat on the edge of the pool, making patterns in the water using my legs for I didn't know how to swim. My friend called out " What are you doing? Come in!" "I don't know how to swim" I replied back. She got out,dripping wet  and sat next to me, ." It's Okay! There is a first time for everything, Give it a try! And see ,the pool doesn't even look that deep" . I stood up and stared at the smaragdine tiles at the bottom of the pool, "Yeah it doesn't look very deep, Maybe my legs will be able to reach the bottom." And the next second, I jumped right in, to realise my legs didn't reach the bottom, I immediately started to panic and could feel myself drowing down, The next thing I know, I was lying on my bed, my panic-stricken friends standing around me. 

So you might be wondering why I titled this as a Physics blog but am saying about an experience which happened 8 years ago. That's because by learning this specific concept ( about which the blog is based on) , I learnt what mistake I made that day. 

'Not everything is what it seems like' ,especially in this peculiar world of Physics. 

As a continuous of my blogs about this vast topic of Light, ( If you haven't read the previous ones, click below! 

The Refractive Indices of Light and its relation with the Snell's Law

More about Refraction of Light through a Rectangular Glass_ Proofs of some basic properties and Factors affecting Refraction 

Why is -v/u = Magnification of Spherical Mirror? ____Proof of Magnification formula  )

We will learn  today about Actual and Apparent Depth 

So...What  are Actual depth and Apparent depth?

 Well, 'Actual Depth' is how deep the tank, pond ( or in my case swimming pool) 'actually' is. Apparent Depth is what the depth seems to be.

You may wonder "Wait, how are there two different depths? What we see is the 'depth' of the object right? (Don't make the same mistake I did 😅 )

This will be true only if the light from the object (bottom of the pool )is travelling in the same medium , as in the case of an empty pool, the only medium there is air. But when the pool is filled, the light has to travel through different mediums, namely water and air. Hence the abnormal phenomenon of light bending occurs -- also famously known as "REFRACTION".

Let's imagine you are standing near the edge of the pool , like this  👇

The light from the bottom of the pool will reach your eyes in this way, it refracts away from the normal (as it goes from denser to rarer medium ) , Now if we extend the light rays we receive backwards,

We realise that, the depth our eyes think it sees is higher than the real depth, but how much higher? Lets figure that out together!

First lets draw triangles along the normal , incident and extended refracted ray. 

 

We get 2 triangles,

ABD drawn along normal and extended refracted ray (AB being apparent depth) 

 and ACE drawn along normal and incident ray (AC being actual depth) 

We also know that ∠DAB = Angle of Refraction ( due to being vertically opposite angles with r)

  and  ∠EAC = Angle of incidence 

By Snell's Law :- 

μ water sin i = μ air sin r 

so, μ 2 (water)  sin ∠EAC = μ1 (air) sin ∠DAB

and by small angle approximation ( visit About Small angle approximation  to know more about it!) 

sin Ө ≈ tan  Ө ≈ Ө

hence 

 μ 2  sin ∠EAC = μ1  sin ∠DAB

 μ 2  DB/ Apparent Depth = μ1 DB/Actual Depth 

By cancelling out DB we get,

And normal shift ( the difference between actual depth and apparant depth ) =

The next time you go to the pool and decide to jump in, I hope you remember this concept and think again especially if you are not a great swimmer ( I just wish there was a time machine  by which I could tell this to my younger self 😅) 

Bonus :-

Now lets just imagine you are underwater ( cough or are Percy Jackson fighting a battle underwater cough ) and you decide to look up and see a bird flying , will the same laws work here too? 

In this situation, (remember, here the light is coming from air to water) 

∠QXY = Angle of refraction ( QX -- Actual Height)

∠PXZ = Angle of Incidence (PX -- Apparent Height)

Hence similar to the previous case , we can say that:-

μAir sin i = μWater sin r (Snell's Law) 

By Finding tan i and tan r , then using small angle approximation we can come to the conclusion:- 

The difference you will find between Apparent-Actual Height and Apparent -Actual Depth is that --

In Depth __ The Apparent Depth is lesser than the Actual Depth

In Height __ The Apparent Height is greater than the Actual Height

and in both the cases you will find that the apparent image is above the original object 

So, the next time you go to the lake or a pond and see a fish swimming , don't forgot to aim low! and when  You and your friends enjoy fish curry don't forgot to remember this blog 😋

That's all folks! This might be the end of this journey , but there are more coming your way from this wonderful concept of Light! So keep your eyes  peeled and minds ready for more blog posts!