Problem 50 - University Physics with Modern Physics

Question

\(\mathrm{A} 45^{\circ}-45^{\circ}-90^{\circ}\) prism is immersed in water. A ray of light is incident normally on one of its shorter faces. What is the minimum index of refraction that the prism must have if this ray is to be totally reflected within the glass at the long face of the prism?

Step-by-Step Solution

Answer

The minimum index of refraction is approximately 1.88.

Step 1: Identify the Critical Angle Condition

Since the ray is incident normally on one face of the prism, it enters the glass without bending. For total internal reflection to occur at the long face, the angle of incidence inside the glass must be greater than or equal to the critical angle, where \( \sin(\theta_c) = \frac{n_{water}}{n_{glass}} \). Here, \( n_{water} = 1.33 \).

Step 2: Determine the Angle of Incidence

Since the refracted angle at the first interface is 0°, the incident angle at the long face will be the angle of the prism itself, which is 45°. Therefore, the condition for total internal reflection requires that \( \sin(45°) \geq \frac{1.33}{n_{glass}} \).

Step 3: Solve for Minimum Index of Refraction

We need \( n_{glass} \) such that \( 1.33 \leq n_{glass} \times \sin(45°) \). Substituting \( \sin(45°) = \frac{\sqrt{2}}{2} \), we have \( n_{glass} \geq \frac{1.33 \times \sqrt{2}}{2} \approx 0.94 \times \sqrt{2} \approx 1.88 \).

Step 4: Conclusion

Thus, the minimum index of refraction of the prism that ensures total internal reflection at its long face when immersed in water is approximately 1.88.

Key Concepts

Prism

A prism is a solid geometric figure with flat surfaces and specific angles between them. In optics, a prism is typically used to bend or refract light. The shape of a prism is important as it determines how light will be refracted or reflected when it enters and exits the prism.

There are different types of prisms, characterized by the angles between their sides. The prism described in the exercise is a right-angled triangular prism with angles of 45°, 45°, and 90°. When light enters this prism, it can undergo refraction and, under the right conditions, total internal reflection. Understanding the geometry of the prism helps us predict how light behaves inside it.

Total Internal Reflection

Total internal reflection is an optical phenomenon that occurs when a light ray travelling in a medium hits the boundary at an angle greater than the critical angle. When this happens, the light does not pass through, but is instead completely reflected back into the medium.

In the exercise, for total internal reflection to occur at the long face of the prism, the light must strike it at an angle equal to or larger than the critical angle. If the conditions are met, the light will not exit through the long face, but will instead be reflected entirely within the prism. This concept is crucial in designing optical devices such as prisms and fiber optic cables.

Index of Refraction

The index of refraction, or refractive index, is a measure of how much a ray of light bends, or refracts, when it enters a medium. Each material has a characteristic refractive index, symbolized by 'n'.

The exercise asks for the minimum index of refraction for the prism to allow total internal reflection. This is calculated by considering the refractive index of water and using the condition at which the sine of the critical angle equals the refractive indices' ratio between two materials (here, the glass prism and water). Without sufficient refractive index, light would not reflect inside the prism and escape from it instead.

Critical Angle

The critical angle is the minimum angle of incidence at which total internal reflection occurs. It is specific to the pair of materials involved and depends on their respective indices of refraction.

In the given problem, the critical angle can be determined by the equation \( \sin(\theta_c) = \frac{n_{water}}{n_{glass}} \). This means that for a light ray travelling from glass to water to be totally reflected inside the glass prism, the angle of incidence must be greater than the determined critical angle. The precise calculation allows us to find the necessary minimum index of refraction to achieve total internal reflection when immersed in water.