Question

Angle of prism is \(\mathrm{A}\) and its one surface is silvered. Light rays falling at an angle at incidence \(2 \mathrm{~A}\) on first surface return back through the same path after suffering reflection at second silvered surface. What is the refractive index of material ? (A) \(\tan \mathrm{A}\) (B) \(2 \sin \mathrm{A}\) (C) \(2 \cos \mathrm{A}\) (D) \(\cos (\mathrm{A} / 2)\)

Short answer

None of the given options is correct.

Step-by-step solution

Identify the angle of incidence, reflection, and refraction

Given that the light rays are falling at an angle \(2A\) on the first surface, the angle of incidence is given by \(i = 2A\). Due to the symmetry of the problem, after reflection, the angle of incidence will be the same \((2A)\) on the second surface. The angle of reflection from the silvered second surface will also be \(2A\), and the angle of refraction will be the same with respect to the first surface \((2A)\).

Apply Snell's law on the first surface

Snell's law states that in a refracting medium: n1 * sin(i) = n2 * sin(r) where n1 and n2 are refractive indices of the two media, and i and r are the angle of incidence and the angle of refraction, respectively. In our case, n1 is the refractive index of air (which is 1), and n2 is the refractive index of the material (which we are trying to find). Therefore, we have: 1 * sin(2A) = n2 * sin(r)

Apply the law of reflection on the silvered surface

Using the law of reflection, which states that the angle of incidence is equal to the angle of reflection, we know that the angle of reflection on the silvered surface is also \(2A\).

Apply Snell's law again on the first surface

Since the light rays return through the same path, the angle of incidence and the angle of refraction on the first surface are still given by \(2A\). Thus, we have: n2 * sin(r) = 1 * sin(2A)

Solve the two Snell's law equations to find the refractive index

From steps 2 and 4, we have two equations: 1 * sin(2A) = n2 * sin(r) (Equation 1) n2 * sin(r) = 1 * sin(2A) (Equation 2) Since both equations are equal to sin(2A), we can equate them: n2 * sin(r) = n2 * sin(r) This indicates that sin(r) is not zero, which means that the light is refracted (not going through the boundary straight). Therefore, we can divide both sides by sin(r). This gives us the refractive index: n2 = sin(2A) / sin(r) Since option (B) gives us the refractive index as n2 = \(2\sin{A}\), we now need to check whether our derived equation matches this option.

Express the angle of refraction in terms of A

As the light rays return back through the same path, the angle of incidence and the angle of refraction on the first surface are equal. Therefore: 2A = r Now, substitute this in the equation from Step 5: n2 = sin(2A) / sin(2A) n2 = 1 This does not match with the given options. However, we made an assumption during the analysis that if the light rays return back through the same path, the angle of incidence and the angle of refraction are equal. This assumption is not necessarily true for all cases. We can therefore choose a different route to find an answer that matches with the given options.

Use symmetry to find the angle inside the prism

Since the angle of reflection is the same on both surfaces, the angle inside the prism will be equal to \(r - A\). This is because both angles add to the large angle of the prism.

Use the sum of the interior angles of a triangle

Since the light rays return back through the same path, the sum of the interior angles should be equal to 180 degrees. Hence, we have: 2A + (r - A) = 180 Simplify: A + r = 180 Now, replace r with 2A from the previous step: A + 2A = 180 3A = 180 A = 60

Rearrange the Snell's law equation to find refractive index

From Equation 1, we can rewrite the Snell's law for the prism: n2 * sin(r) = sin(2A) Replace A with 60 and r with 2A: n2 * sin(120) = sin(120) Since sin(120) ≠ 0, divide both sides by sin(120): n2 = 1 This result still does not match any of the given options, which indicates that there may be a problem with the problem statement or the given options. So none of the given options is correct.