# Prove of refractive index

#### primarygun

Can anyone show me a proof of "refractive index= (real depth)\(apparent depth) "?
I found the proof in my book has a mistakes and I found some contradicts to this equation.

Last edited:
Related Introductory Physics Homework Help News on Phys.org

#### Pengwuino

Gold Member
Why do you say there are mistakes and contradictions?

#### primarygun

The book said used a pair of similar triangles to infer it but the triangles are not similar.
Do you want a picture? If you want, I can upload it now.

#### primarygun

Hope it helps you bring me out of the troubles.

#### Pengwuino

Gold Member
Ugh, basic geometry :(

I think they are considered similar triangles.

is it correct?

#### Pengwuino

Gold Member
yah im pretty sure it is... hopefuly someone will verify though... its 3am here :D

#### primarygun

But, I think the equation should be correct as I saw it in many books.
We need the someone's help;

#### HallsofIvy

Homework Helper
primarygun said:
Can anyone show me a proof of "refractive index= (real depth)\(apparent depth) "?
I found the proof in my book has a mistakes and I found some contradicts to this equation.
What DEFINITION of 'refractive index' does your book give? (It is possible to use "refractive index= (real depth)/(apparent depth)" as the definition.)

#### primarygun

index of refraction.

Pengwuino said:
yah im pretty sure it is... hopefuly someone will verify though... its 3am here :D
if the two triangles were similar, wouldn't r = i, or 90 degrees - r = i?

not sure if giving the ratio of the sines will show this. :grumpy:

#### sniffer

i derived this equation before. the formula is just an approximation for small angle case, where $sin\theta=tan\theta$ for small angle. i.e. it is for the case where you almost look vertically down to the object from above. you can derive it easily by drawing slender triangles. very easy.

if you get stuck i can post a proper solution with the drawing.

#### Gokul43201

Staff Emeritus
Gold Member
The triangles are NOT similar. That derivation is just extremely poorly worded (or was written by someone who knew the result and "made up" the proof). What it is actually using is the small angle approximation that sniffer mentions.

Anyway, I believe Primary's doubts have long been resolved.

#### Nerro

According to the definition these triangles áre similar because when you shrink one side the become congruent which is apparently enough to qualify them as "similar" (I have looked this up.).

Frankly I thought the explanation was pretty straightforward...

Last edited:

### Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving