Chapter 9: Problem 32

Would the critical angle for a glass-water interface be less than. equal to, or greater than the critical angle for a glass-air interface? Explain your choice.

Short Answer

Expert verified

Answer: The critical angle for a glass-water interface is greater than the critical angle for a glass-air interface.

Step by step solution

Define critical angle.

The critical angle is the angle of incidence at which light is refracted to an angle of 90 degrees from the normal. It's expressed with the following formula: $$\sin(\theta_c) = \frac{n_2}{n_1}$$ where $\theta_c$ is the critical angle, $n_1$ is the refractive index of the first medium, and $n_2$ is the refractive index of the second medium.

Identify the known values.

We know the refractive indices for the three materials in question: glass, water, and air. - The refractive index of glass is approximately 1.5. - The refractive index of water is approximately 1.33. - The refractive index of air is approximately 1.00.

Calculate the critical angle for a glass-water interface.

Using the formula above, we can calculate the critical angle for light going from glass to water. $$\sin(\theta_{c_{gw}}) = \frac{n_{water}}{n_{glass}}$$ $$\theta_{c_{gw}} = \arcsin\left(\frac{1.33}{1.5}\right)$$ $$\theta_{c_{gw}} \approx 61.03^{\circ}$$

Calculate the critical angle for a glass-air interface.

Similarly, we can determine the critical angle for light going from glass to air. $$\sin(\theta_{c_{ga}}) = \frac{n_{air}}{n_{glass}}$$ $$\theta_{c_{ga}} = \arcsin\left(\frac{1.00}{1.5}\right)$$ $$\theta_{c_{ga}} \approx 41.81^{\circ}$$

Compare the critical angles.

Now that we have calculated the critical angles for both interfaces, we can compare them. - Critical angle for glass-water interface: $\theta_{c_{gw}} \approx 61.03^{\circ}$ - Critical angle for glass-air interface: $\theta_{c_{ga}} \approx 41.81^{\circ}$ Since $61.03^{\circ}$ is greater than $41.81^{\circ}$, the critical angle for a glass-water interface is greater than the critical angle for a glass-air interface.

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Would the critical angle for a glass-water interface be less than. equal to, or greater than the critical angle for a glass-air interface? Explain your choice.

Short Answer

Step by step solution

Define critical angle.

Identify the known values.

Calculate the critical angle for a glass-water interface.

Calculate the critical angle for a glass-air interface.

Compare the critical angles.

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