Using a double angle formula with Snell's Law

[smeiste]

Homework Statement

A light ray is incident from air onto a glass surface with an index of refraction n = 1.56. Find the angle of incidence for which the corresponding angle of refraction is one-half the angle of incidence. Both angles are defined with the normal to the surface.

Homework Equations

n = sin a1/sin a2 (Snell's Law)

sin(a + b) = (sin a x cos b) + (cos a x sinb)

sin(2a) = 2 sin a cos a

The Attempt at a Solution

Somehow you plug in the double angle formula into snells law. but I don't understand how this is a down. a breakdown of the equations used to solve problem would be extremely helpful :)

 

[ehild]

The angle of incidence is 2a, the angle of refraction is a. Plug into Snell's law.

ehild

 

[smeiste]

but then how do you solve for a?

 

[ehild]

Show your equation.

ehild

 

[smeiste]

sin(a)/sin(2a) = 1.56

 

[gneill]

sin(a)/sin(2a) = 1.56

What happens if you apply the sine double angle formula here?

 

[bobquantum]

Your equation is actually incorrect. by Snell's formula,
sin a = 1.56Apply the duble angle formula to sin (2a) to get 2sin(a)cos(a). since a does not equal an integral multiple of pi, you can divide out sin(a). From there, you can find that cos(a) = 0.321
Use the inverse of cosine function to solve for a = 71.306 degrees.

 

[smeiste]

if sin a = 1.56, how do you apply the double angle formula?

 

[gneill]

Your equation is actually incorrect. by Snell's formula,
sin a = 1.56Apply the duble angle formula to sin (2a) to get 2sin(a)cos(a). since a does not equal an integral multiple of pi, you can divide out sin(a). From there, you can find that cos(a) = 0.321
Use the inverse of cosine function to solve for a = 71.306 degrees.

Snell's Law:
$$\frac{sin(\theta 1)}{sin(\theta 2)} = \frac{v1}{v2} = \frac{n2}{n1}$$

 

[smeiste]

im so confused.. could someone just show the steps on how to solve it? i think much better in equations than words.

 

[gneill]

Take Snell's law as I wrote it above (You can ignore the velocity ratio, I only included it for completeness. It's not needed for this problem). Substitute the given values for the angles and indexes of refraction. What do you get?

 

[smeiste]

isnt that what i wrote before? sin a/sin 2a = 1.56/1.

 

[smeiste]

sorry, i see. sin2a/sina = 1.56/1

 

[smeiste]

ah ha! so sin 2a = 2sinacosa and the sins cancel out, giving you a final equation of cos a = 0.78. thanks very much everyone!