A triangular glass prism with apex angle 60.0^∘ has an index of refraction of 1.50 . (a) Show th

A triangular glass prism with apex angle 60.0^∘ has an index...

A triangular glass prism with apex angle $60.0^{\circ}$ has an index of refraction of $1.50 .$ (a) Show that if its angle of incidence on the first surface is $\theta_{1}=48.6^{\circ},$ light will pass symmetrically through the prism as shown in Figure $35.17 .$ (b) Find the angle of deviation $\delta_{\min }$ for $\theta_{1}=48.6^{\circ} .$ (c) What If? Find the angle of deviation if the angle of incidence on the first surface is $45.6^{\circ} .$ (d) Find the angle of deviation if $\theta_{1}=51.6^{\circ} .$

Key Concepts

Minimum Deviation

Minimum deviation is the smallest possible angle by which a light ray is deflected when it traverses a prism. This phenomenon occurs under the condition of symmetric passage, when the incident and emergent angles are equal. The concept of minimum deviation is not only useful for characterizing the behavior of light in prisms but also provides a practical method for determining the refractive index of the prism material through measured angles.

Symmetric Passage through a Prism

A symmetric passage occurs when a light ray enters a prism in such a way that the internal angles of incidence and emergence are equal. This condition typically results in a minimum deviation of the light ray. In practice, symmetric passage simplifies the analysis, since the light experiences equal bending at each interface of the prism, and is a common setup for accurately measuring optical properties like the refractive index.

Geometry of Prisms

The geometry of a prism, particularly the apex angle, directly influences the path that light takes through it. By understanding the relationships between the internal angles within the prism and the angles of incidence and emergence, one can derive formulas to predict how the light will bend. These geometric relationships are crucial for establishing conditions such as symmetric paths inside the prism.

Snell's Law

Snell's Law is the fundamental principle that governs the bending of light when it passes from one medium to another. It states that the product of the refractive index and the sine of the angle of incidence in the first medium is equal to the product of the refractive index and the sine of the angle of refraction in the second medium. This law is essential for determining the paths that light rays take when interacting with different optical surfaces, such as those in a prism.

Transcript

00:01 We have been given a prism with an apex angle of 60 degrees and a refractive index of n2, which is equal to 1 .5.

00:15 Now we have an angle of entry of a ray into this prism at 48 .6 degrees, and we're being asked to prove that the ray is symmetrical, as shown in the image that they have provided for us.

00:31 So this is a 60 degree angle.

00:34 This is our ray.

00:36 And of course, this is going to be symmetric and normal to the bottom.

00:47 So we have an angle entering at, what, 48 .6 degrees.

00:52 So this is theta 1.

00:53 And this is the normal.

00:56 No, sorry.

01:00 This is the normal to the surface.

01:04 And this will be exiting at the exact same angle.

01:09 So we want to prove that this angle right here, we want to prove that this angle is the same as this one.

01:24 So to do that, we're going to have to do some magic.

01:32 So we are going to say that n1, sine theta, 1 is equal to n2 sine theta 2.

01:43 And since we know that the outside medium is air, we know that's 1.

01:48 So we're going to want to find angle 2, which is located right.

02:05 So our theta 2 is located right here.

02:11 So we want to determine what that is.

02:14 So to do that, we're going to take sign of theta 1, divided by n2, and then find the inverse sign of that whole situation, which will give us theta 2.

02:26 Now that will give us sign of 48 .6 over 1 .5 equal to theta 2.

02:36 Theta 2 will then be equal to 30 degrees.

02:50 So we have now determined that theta 2 is 30 degrees.

02:55 We know that this is 60 degrees.

02:58 So if this guy is 30 degrees, and this isn't normal, that means that this angle, which we're going to call alpha, right, alpha plus theta 2 is going to have to equal 90 degrees, which means that this angle right here, alpha, since this is 30 degrees, alpha is going to be 60 degrees.

03:22 So that means if we have 260 degree angles, this one also has to be 60 degrees.

03:27 That's just basic geometry.

03:30 All the angles have to add up to 180 degrees.

03:32 So now we determine that this angle is also 60 degrees.

03:36 So i'm going to make this a little more clear.

03:38 Since we know that this angle is 60 degrees, that means that this angle and this angle, which we're going to call theta 3, have to equal, you guessed it, they have to equal 90 degrees...

Please give Ace some feedback