Understanding Snell's Law: Common Mistakes and Troubleshooting Tips

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In summary, the problem involves finding the angle of incidence at the exit point of a ray experiencing total internal reflection. The equations used are n1sinθ1=n2sinθ2 and θ2 = arcsin(n1/n2). However, there is a discrepancy in the equations and the angle θ2 should be calculated as 90° - θ2 = arcsin(n1/n2). This leads to the conclusion that the angle of incidence must be greater than the critical angle in order for internal reflection to occur.
  • #1
jegues
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Homework Statement



See figure attached for problem statement.

Homework Equations





The Attempt at a Solution



I'm getting a disagreement in my equations so I must be doing something wrong.

We have two unknowns,

[tex]n_{1},\theta_{2}[/tex]

My first equation,

[tex]n_{1}sin\theta_{1}=n_{2}sin\theta_{2}[/tex]

my second equation (This is where I think I am misunderstanding something)

Since it is experiencing total internal reflection,

[tex]\theta_{2} = arcsin\frac{n_{1}}{n_{2}}[/tex]

Or in other words,

[tex]sin\theta_{2} = \frac{n_{1}}{n_{2}}[/tex]

If I plug this into my first equation I get an inconsistency because,

[tex]sin\theta_{1} \neq 1[/tex]

What am I doing wrong?
 

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  • #2
Find the angle of incidence at the exit point of the ray. That should be greater than the critical angle so that internal reflection happen.

ehild
 
  • #3
jegues said:

Homework Statement



See figure attached for problem statement.

Homework Equations





The Attempt at a Solution



I'm getting a disagreement in my equations so I must be doing something wrong.

We have two unknowns,

[tex]n_{1},\theta_{2}[/tex]

My first equation,

[tex]n_{1}sin\theta_{1}=n_{2}sin\theta_{2}[/tex]

my second equation (This is where I think I am misunderstanding something)

Since it is experiencing total internal reflection,

[tex]\theta_{2} = arcsin\frac{n_{1}}{n_{2}}[/tex]  This should be:  90° ‒ θ2= arcsin(n1/n2),
which is: sin(90° ‒ θ2) = cos(θ2) = (n1/n2)


Or in other words,

[tex]sin\theta_{2} = \frac{n_{1}}{n_{2}}[/tex]

If I plug this into my first equation I get an inconsistency because,

[tex]sin\theta_{1} \neq 1[/tex]

What am I doing wrong?

See comment in red above.
 
  • #4
SammyS said:
See comment in red above.

Thank you.
 
  • #5


Hi there,

It seems like you might have misunderstood the concept of total internal reflection. In this scenario, the angle of incidence (theta_1) is greater than the critical angle, meaning that all of the light is reflected back into the medium of higher refractive index (n_1). This means that the angle of refraction (theta_2) is actually 90 degrees, since the light never actually enters the medium of lower refractive index (n_2). Therefore, your second equation should be:

n_1sin(theta_1) = n_2sin(90)

This should resolve the inconsistency you were experiencing. Keep in mind that total internal reflection only occurs when the angle of incidence is greater than the critical angle, and in this case, the angle of refraction will always be 90 degrees. I hope this helps clarify things for you. Keep up the good work!
 

What is Snell's Law?

Snell's Law, also known as the Law of Refraction, is a formula that describes the relationship between the angles of incidence and refraction of light as it passes through the boundary between two different mediums, such as air and water.

Who discovered Snell's Law?

Snell's Law was discovered by Dutch mathematician and astronomer Willebrord Snellius in 1621.

What is the mathematical formula for Snell's Law?

The mathematical formula for Snell's Law is n1*sin(theta1) = n2*sin(theta2), where n1 and n2 are the indices of refraction of the two mediums, and theta1 and theta2 are the angles of incidence and refraction, respectively.

What is the significance of Snell's Law?

Snell's Law is significant because it explains how light changes direction when it passes through different mediums, and it is also used to calculate the bending of light when it passes through lenses, prisms, and other optical devices.

Can Snell's Law be applied to other types of waves besides light?

Yes, Snell's Law can be applied to any type of wave that travels through different mediums, such as sound waves, water waves, and seismic waves.

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