Introduction:
When a ray of light passes through a prism, it undergoes refraction, which causes it to change direction. The amount of deviation depends on the refractive index of the material of the prism. In this case, we have a right isosceles prism that deviates the ray of light by 90 degrees. We need to determine the minimum refractive index of the prism's material to achieve this deviation.

Understanding refraction:
Refraction occurs when light passes from one medium to another, and its speed changes. The change in speed causes the light ray to bend. The amount of bending depends on the refractive indices of the two media involved. The refractive index (n) of a material is the ratio of the speed of light in a vacuum (c) to the speed of light in that material (v): n = c/v.

Deviation by a right isosceles prism:
A right isosceles prism is a prism with a right angle between its two equal faces. When a ray of light enters such a prism perpendicular to one of its faces, it undergoes two refractions and one total internal reflection. The deviation caused by this prism depends on the refractive index of the material.

Minimum refractive index for 90-degree deviation:
To achieve a 90-degree deviation, the incoming ray of light should be incident at the critical angle. The critical angle (θc) is the angle of incidence at which the refracted ray travels along the interface between the two media. For a right isosceles prism, the critical angle occurs at the interface between the prism and air.

Calculating the minimum refractive index:
The critical angle (θc) can be calculated using the formula: sin(θc) = 1/n, where n is the refractive index of the prism's material. For a 90-degree deviation, the critical angle is 45 degrees.

To find the minimum refractive index, we can rearrange the formula as: n = 1/sin(θc).

Using the critical angle of 45 degrees, we can substitute it into the equation:
n = 1/sin(45°) = 1/√2 ≈ 0.707.

Therefore, the minimum refractive index of the prism's material should be approximately 0.707 to achieve a 90-degree deviation.

Conclusion:
In conclusion, for a ray of light to be deviated by 90 degrees in a right isosceles prism, the minimum refractive index of the prism's material should be approximately 0.707. This can be achieved by ensuring the incident angle of the ray is at the critical angle, which is 45 degrees in this case.