A ring with a colorless gemstone is dropped into water. The gemstone becomes invisible when submerged. Can it be a diamond? Explain.

Refractive index (n) is a measure of how much a substance slows down light passing through it, causing the light to bend. The greater the refractive index, the slower light moves through the substance, and therefore the more bending or refraction occurs. When light passes between media with different refractive indices, a characteristic called total internal reflection might occur.

Total internal reflection occurs when light travels from a medium with a higher refractive index (n1) to a medium with a lower refractive index (n2) and hits the boundary between the two at an angle larger than the critical angle. In this situation, the light is reflected back into the higher-index material instead of transmitting through the boundary. The critical angle can be calculated by using Snell's law for refraction, which is given by \( n_{1} \times sin(\theta_{1}) = n_{2} \times sin(\theta_{2}) \) Here, \(\theta_{1}\) and \(\theta_{2}\) are the angles the light rays make with the normal (perpendicular) to the boundary between the two media.

To determine if the gemstone can be a diamond, we need to know the refractive indices of diamond and water. The refractive index of diamond is approximately 2.42, while that of water is approximately 1.33.

Now that we have the refractive indices of diamond and water, let's check if total internal reflection occurs. Applying Snell's law for the two media obtained in Step 3: \( 2.42 \times sin(\theta_{1}) = 1.33 \times sin(\theta_{2}) \) \( sin(\theta_{1}) = 1.33 / 2.42 \times sin(\theta_{2}) \) By calculating the critical angle, we have \( \theta_{1} \approx 41.56^{\circ} \).

As we have calculated the critical angle between diamond and water, we can determine if the gemstone can be a diamond. When the gemstone is submerged in water, if the light entering the diamond from water hits the diamond-water boundary at an angle larger than the critical angle (41.56 degrees), total internal reflection will occur, and the gemstone will become invisible. However, considering the gemstone's size and shape, it is highly unlikely that all light entering would hit the boundary at an angle larger than the critical angle. Thus, the gemstone will not become entirely invisible, and the observed phenomenon is not consistent with a diamond. In conclusion, it is unlikely that the gemstone in question is a diamond, as a diamond in water would not exhibit total invisibility due to its refractive index and the critical angle for total internal reflection with water.