Question

the critical angle at the boundary of material x and material y is found to be 57.1 degrees. if the index of refraction for material y is 1.41, what is the index of refraction of material x?

total internal reflection cannot occur.
1.68
1.47
1.84

Explanation:

Step1: Recall the formula for critical angle

The formula for critical angle \( \theta_{c}\) is \( \sin\theta_{c}=\frac{n_{Y}}{n_{X}}\), where \(n_{X}\) is the refractive index of material \(X\) (optically denser medium) and \(n_{Y}\) is the refractive index of material \(Y\) (optically rarer medium).

Step2: Rearrange the formula to solve for \(n_{X}\)

From \( \sin\theta_{c}=\frac{n_{Y}}{n_{X}}\), we can get \(n_{X}=\frac{n_{Y}}{\sin\theta_{c}}\).

Step3: Substitute the given values

Given \(n_{Y} = 1.41\) and \( \theta_{c}=57.1^{\circ}\), then \(n_{X}=\frac{1.41}{\sin(57.1^{\circ})}\).
We know that \( \sin(57.1^{\circ})\approx0.84\).
So \(n_{X}=\frac{1.41}{0.84}=1.68\).

Answer:

1.68