Question

A collimated laser beam strikes the left side (A) of a glass block at an angle of \(20.0^{\circ}\) with respect to the horizontal, as shown in the figure. The block has an index of refraction of 1.55 and is surrounded by air, with an index of refraction of \(1.00 .\) The left side of the glass block is vertical \(\left(90.0^{\circ}\right.\) from horizontal) while the right side \((\mathrm{B})\) is at an angle of \(60.0^{\circ}\) from the horizontal. Determine the angle \(\theta_{\mathrm{BT}}\) with respect to the horizontal at which the light exits surface \(B\).

Short answer

Answer: To find the angle at which the light exits the glass block on the right side (\(\theta_{BT}\)), follow these steps: 1. Calculate the angle of refraction at point A (\(\theta_2\)) using Snell's Law. 2. Determine the angle of incidence at point B (\(\theta_{BI}\)) by subtracting \(\theta_2\) from 60 degrees. 3. Apply Snell's Law again at point B to find the angle at which the light exits the glass block (\(\theta_{BT}\)). 4. Plug in the value for \(\theta_2\) obtained in step 1 into the equation \(\theta_{BT} = \arcsin\left[1.55 \cdot \sin(60^{\circ} - \theta_2)\right]\) to get the final answer for \(\theta_{BT}\).

Step-by-step solution

Calculate the angle of refraction at point A

First, we'll apply Snell's Law at point A. Snell's law states: \(n_1 \cdot \sin \theta_1 = n_2 \cdot \sin \theta_2\) Where \(n_1\) is the index of refraction of the first medium (air in this case), \(n_2\) is the index of refraction of the second medium (glass), and \(\theta_1\) and \(\theta_2\) are the angles of incidence and refraction, respectively. From the problem, we have: \(n_1 = 1.00\) \(\theta_1 = 20^{\circ}\) \(n_2 = 1.55\) Now, we can solve for \(\theta_2\): \(\sin \theta_2 = \frac{n_1 \cdot \sin \theta_1}{n_2}\) \(\sin \theta_2 = \frac{1.00 \cdot \sin 20^{\circ}}{1.55}\)

Calculate the angle of incidence at point B

We must now find the angle of incidence at point B. From the given geometry, surface B is inclined at \(60^{\circ}\). Therefore, the angle of incidence at B can be found by subtracting the angle of refraction at A (\(\theta_2\)) from 60 degrees: \(\theta_{BI} = 60^{\circ} - \theta_2\)

Apply Snell's Law at point B

We will now apply Snell's Law again at point B. This time, the light travels from the glass to the air. Thus: \(n_3 = 1.00\) \(\theta_3 = \theta_{BT}\) (the angle we're looking for) Now, we can solve for \(\theta_3\): \(\sin \theta_3 = \frac{n_2 \cdot \sin \theta_{BI}}{n_3}\) \(\sin \theta_3 = \sin(\theta_{BI}) \cdot 1.55\)

Calculate the angle \(\theta_{BT}\)

Now, with the obtained expression for \(\sin \theta_3\), we can find the angle \(\theta_{BT} = \theta_3\): \(\theta_{BT} = \arcsin\left[1.55 \cdot \sin(60^{\circ} - \theta_2)\right]\) Plug in the value for \(\theta_2\) obtained in step 1 to get the final answer for \(\theta_{BT}\).

Key concepts

Refraction

Refraction is a fascinating natural phenomenon that occurs when light waves pass from one medium to another with a different density, such as from air into water. This causes the light to change its speed and direction. Imagine you were pushing a cart from a smooth sidewalk onto the grass; the wheels slow down unevenly, causing the cart's path to bend. This is quite similar to how light bends, or 'refracts', when it enters a new medium at an angle.

Why does this happen? It's because light travels at different speeds in different materials. In materials with a higher 'optical density', such as glass or water, light slows down. If the light hits the surface between these materials at a slant, one side of the light wave will slow down before the other, resulting in a change in the light's direction. This bending of light between two materials with different densities is what we call refraction.

Index of Refraction

The index of refraction, often represented by the symbol 'n', quantifies how much the light bends when it enters a medium. It is defined as the ratio of the speed of light in a vacuum, which is the fastest speed at which light can travel, to the speed of light in the medium in question. The index of refraction has no units—it's a ratio, after all! For most purposes, we consider the index of refraction of a vacuum to be exactly 1, and so the values for all other materials are compared against this.

The higher the index, the slower light travels in that medium and the more it bends, or refracts. For example, typical glass has an index of refraction around 1.5, meaning light travels at roughly two-thirds the speed in glass as it does in a vacuum (or air). This property plays a key role in designing lenses for glasses, cameras, and even the microscopes we use in science labs.

Angle of Incidence

The angle of incidence is at the core of understanding how refraction works. It is the angle between the incoming light ray and an imaginary line perpendicular to the surface, called the 'normal'. To visualize this, think of the normal as a standing flagpole's shadow on the ground, and the angle of incidence as the angle you would tilt a straight stick from that shadow.

When measuring this angle, it's important to note that it's not relative to the surface but to the normal. This is why even if a surface is slanted, like the side of a glass prism, you can still calculate the angle of incidence by measuring how much the incoming light ray diverges from the perpendicular normal. The angle of incidence is crucial in determining how much the light ray will bend when it goes from one medium to another, which is predicted by Snell's Law.

Angle of Refraction

Mirroring the concept of the angle of incidence, the angle of refraction describes the angle between the refracted light ray and the normal within the new medium. Continuing the analogy used for the angle of incidence, if you push that same stick into water at an angle, the angle of refraction would be the tilt of the stick's submerged part relative to the water’s surface's normal.

The angle of refraction is influenced by the material's index of refraction. If light is moving from a less optically dense medium to a more dense one, like from air to water, it slows down and the angle of refraction is smaller than the angle of incidence — the path of the light bends towards the normal. Conversely, transitioning from a denser to a less dense medium causes light to speed up, creating a larger refraction angle as the light bends away from the normal. By understanding the angle of refraction, one can gain insight into the behavior of light as it transitions between various media, which is vital for applications ranging from eyewear to fiber optic communications.