Question

bridging problem: reflection and refraction in (figure 1) shown a rectangular glass block that has a metal reflector on one face and water on an adjoining face. a light beam strikes the reflector as shown. you gradually increase the angle \\( \theta \\) of the light beam. if \\( \theta \geq 51.2^\circ \\), no light enters the water. what is the speed of light in this glass? figure 1 of 1 light beam glass reflector water interface; that is, there must be total internal reflection. part b the target variable is the speed of light \\( v \\) in the glass, which you can determine from the index of refraction \\( n \\) of the glass. which equations will you use to find \\( n \\) and \\( v \\)? check all that apply. view available hint(s) \\( \sin \theta_a = n_b \sin \theta_b \\) \\( \sin \theta_{crit} = \frac{n_b}{n_a} \\) \\( v = \frac{c}{n} \\) \\( \lambda = \frac{\lambda_0}{n} \\) \\( \theta_r = \theta_a \\) \\( \tan \theta_p = \frac{n_b}{n_a} \\) submit previous answers incorrect; try again; one attempt remaining

Explanation:

Step1: Analyze Total Internal Reflection Condition

For total internal reflection at the glass - water interface, the light is going from glass (medium \(a\)) to water (medium \(b\)). The critical angle \(\theta_{crit}\) satisfies \(\sin\theta_{crit}=\frac{n_b}{n_a}\) (since \(n_a>n_b\) for total internal reflection to be possible). Also, from Snell's law, when the light is incident on the reflector, after reflection (angle of reflection \(\theta_r = \theta_a\) because of the law of reflection), and then at the glass - water interface, we have the condition for total internal reflection. The formula for the speed of light in a medium is \(v = \frac{c}{n}\), where \(c\) is the speed of light in vacuum and \(n\) is the refractive index of the medium.

Step2: Evaluate Each Option

• \(\sin\theta_a=n_b\sin\theta_b\): This is Snell's law, but in the context of total internal reflection at the glass - water interface, we are dealing with the critical angle condition, not the general Snell's law for refraction (since no light is refracting out, it's total internal reflection). So this is not applicable.
• \(\sin\theta_{crit}=\frac{n_b}{n_a}\): This is the correct formula for the critical angle when going from medium \(a\) (glass) to medium \(b\) (water) (\(n_a>n_b\)). This is relevant for the total internal reflection condition given (\(\theta\geq51.2^{\circ}\) which is the critical angle or more).
• \(v = \frac{c}{n}\): This is the formula for the speed of light in a medium with refractive index \(n\). Since we need to find the speed of light in glass, this formula is relevant as we can find \(n\) from the critical angle condition and then use this formula to find \(v\).
• \(\lambda=\frac{\lambda_0}{n}\): This formula is about the wavelength of light in a medium, which is not relevant for finding the speed of light in the medium in this context.
• \(\theta_r=\theta_a\): This is the law of reflection (angle of reflection equals angle of incidence). Since the light beam strikes the reflector, this law is applicable as the reflection at the metal reflector will follow this law, which is part of the path of the light beam before reaching the glass - water interface.
• \(\tan\theta_p=\frac{n_b}{n_a}\): This is the formula for the polarizing angle (Brewster's angle), which is not relevant for total internal reflection or finding the speed of light in the medium.

Answer:

The correct options are:

• \(\sin\theta_{crit}=\frac{n_b}{n_a}\)
• \(v = \frac{c}{n}\)
• \(\theta_r=\theta_a\)