Finding the angle of refraction

[RabbitWho]

Homework Statement

The refractive index of diamond is 2.42. A ray of light passing from the inside of a diamond into air makes an angle of incidence of 20 degrees. Find the angle of refraction.

Homework Equations

n = refractive index
aNd = 2.42 (my book said the speed of light in a vacuum and in air are close enough to treat them as the same for our purposes)
dNa = 1/aNd
Sin i / Sin r = dNa
i = 20 degrees

The Attempt at a Solution

0.413 = dNa
Sin 20 degrees = 0.3420
0.3420 / sin r = 0.413
therefore 0.3420 x 0.413 = sin r
sin r = 0.142
arc sin of 0.142 = 8.159 degrees

This is wrong, the answer should be 55.9 degrees..

Where did I go wrong?
Did I go wrong in my maths or in my use of the formula?

If someone could point me in the right direction that would be wonderful.

 

[Simon Bridge]

I think you los track of the refractive index ration and which to use where.
Instead use Snell's law as she is writ:
$\qquad n_1\sin\theta_1 = n_2\sin\theta_2$
... that way you can keep track.

Since the ray passes from the diamond into the air, which way will it bend (towards the normal or away from the normal)?

 

[kuruman]

therefore 0.3420 x 0.413 = sin r

Why is this correct?

Start with $n \sin \theta_i = \sin \theta_r$ and solve for $\sin \theta_r$.

 

[RabbitWho]

I think you los track of the refractive index ration and which to use where.
Instead use Snell's law as she is writ:
$\qquad n_1\sin\theta_1 = n_2\sin\theta_2$
... that way you can keep track.

Since the ray passes from the diamond into the air, which way will it bend (towards the normal or away from the normal)?

That formula isn't taught in my book. I saw them explain it on Kahn Academy but I don't really understand it. I think it is just another way to express the same thing. I have to stick with the formula given for my exam. If I use any formula which is not on the syllabus there is a good chance I will be marked wrong, depending on who corrects my test. These are the formula:

Snell's law: n = sin i / sin r
_xn_y = 1 / _yn_x

It will be bent away from the normal.

Why is this correct?

Start with $n \sin \theta_i = \sin \theta_r$ and solve for $\sin \theta_r$.

Why isn't it correct? The formula says sin i over sin r = dNa
So I have to multiply sin i by dNa to get sin r, right? Is my algebra wrong again?

 

[kuruman]

OK, we will go by the formula that you are familiar with. I understand sin i is the sine of the angle of incidence and that sin r is the sine of the angle of refraction. Please explain what dNa stands for because I am unfamiliar with the notation.

 

[RabbitWho]

Sorry about that!
I should have realized it wouldn't be clear.

we are using n for refractive index as well, but
aNd (_an_d)
d stands for diamond and a stands for air, so aNd is the refractive index of light traveling from air into a diamond and dNa is the refractive index of light traveling from a diamond into air.

 

[kuruman]

And, as the problem states, the refractive index of diamond is 2.42, which one do you think is given, _dn_a or _an_d?

 

[RabbitWho]

it has to be _an_d, doesn't it? Because all it says is "the refractive index", and the normal one to give is light traveling from air (or a vacuum) into the medium.

 

[kuruman]

OK. Then, if light goes from air to diamond, Snell's law is
$$ \frac{sin \theta_{air}}{sin \theta_{diamond}} = {_an_d} $$
Right?
And when light goes from diamond into air,
$$ \frac{sin \theta_{diamond}}{sin \theta_{air}} = {_dn_a} $$
Right?
Can you solve for what you need? Note that these two equations say that it doesn't matter what is "incident" and what is "refracted". When you send light back where it came from, it will retrace its path. Also note that I used subscripts "diamond" and "air" instead of "i" and "r" to avoid confusion of what's what. I recommend you do the same if you are to use the form of Snell's law that was given to you. The rule is this "The subscript of the sine in the numerator matches the first subscript of n and the subscript of the sine in the denominator matches the second subscript of n."

 

[RabbitWho]

Ah! Thank you, that was a silly mistake. Let me see if I can do it now

0.413 = dNa
Sin 20 degrees = 0.3420
sin r [sin theta diamond] / 0.3420 = 0.413

I think I am still going wrong somewhere because I can't see how to get 55.9 degrees out of what I have Sorry about this

 

[kuruman]

0.413 = dNa

OK

Sin 20 degrees = 0.3420

OK
Use the two OK'ed numbers above and forget the others. Start with $\frac{\sin \theta_{diamond}}{ \sin \theta_{air}}={_dn_a}$
Which one of the quantities in the equation are you looking for and how can you find it?

 

[Simon Bridge]

Wow - no wonder you are confused. So besides what kuruman is telling you ...
If the speed of light in the medium is v, and the speed of light in vacuum is c, then, by definition, the refractive index for the material is n=c/v
Snells law is for light passing through an interface between two arbitrary transparent materials.
I wanted to point this out because it will help you later when you talk to people outside your particular course.

Refractive index for air is usually treated as 1 ... so for light passing from air to a material, $\sin\theta_i = n\sin\theta r$ and for light passing from the material to the air, $n\sin\theta_i = \sin\theta_r$ ... these are mathematically the same equation.
So the ratio $\sin\theta_i/\sin\theta_r$ is 1/n, if light goes from air to medium, and n if the light goes from medium to air.

The genaral Snells law is better because you only have one equation to get straight.
The maths does not care which direction the light travels, so there's one thing you don't need to worry about, and those mNa and aNm things take care of themselves so there's another two things not to worry about. The downside is that you need to be able to do algebra.
I'm pretty bad at memorizing stuff, so I always feel it is better to reduce the amount of memorization needed to do anything.

I'm going to let kuruman help you out here though... you don't need two people talking at you.