Mathematics of refraction of light

[anantchowdhary]

Can anyone please prove that if d1=EF and if d2=EG

and angle MEZ=HEF=angle of incidence

and angle HEG = angle of refraction.

Please! can somebody help me prove that d1/d2=refractive index

taking sin i/sin r=refractive index

(The glass slab is ABCD and is kept in vacuum)

 

[berkeman]

As you know, you must show your own work in order for us to help you. This seems pretty straightforward using Snell's law and a little trig. Please set up the equations, and we'll see if you are on the right track.

 

[anantchowdhary]

Hey I am sorry but with trig I am getting some equations but i don't think I am on the correct path:

d1=sin i{HF} ...(i)
d2=sin r{HG} ...(ii)

(i) / (ii)

d1/d2=sini.(HG+GF)/sinr(HG)
therefore

d1/d2=mu .(HG+GF)/(HG)

I get stuck after this.Help will be appreciated

 

[berkeman]

I only looked at it briefly, but I'd start with using cos() since EH is the same for both. I'll try to stop by again later if I can and spend more time looking at it.


EDIT: fixed my typo EF-->EH

 

[anantchowdhary]

after cos how do ii convert it in terms of sin?

 

[berkeman]

Okay, I'm at home now, and back online for a bit. If you haven't figured it out yet, here are a couple hints.

Since cos() is more useful for describing the physical situation, but Snell's Law is usually written with sin(), then the logical thing to try using would be the combination tan(), right? So start with these:

$$tan(\theta_i) = \frac{sin\theta_i}{cos\theta_i} = \frac{FH}{EH}$$

$$tan(\theta_r) = \frac{sin\theta_r}{cos\theta_r} = \frac{GH}{EH}$$

Try using that to see if that gets you to the solution. BTW, when you look at the drawing of the refracted ray, you see how raising the n will pull in the $$\theta_r$$ and shorten d2? Visualizing what happens as you change a variable in a problem can help you gain the intuition to help you solve problems. In this case, the relationship direction is intuitive, but it takes doing the math to show that the ratio is just the d1/d2 ratio.

 

[anantchowdhary]

Hey I am grateful to you for the help provided.But I am sorry i just don't seem to get the desired result.Also,

Experimentally i have verified that d1/d2=mu

But I am sorry I am not able to do it mathematically