(II) When a Newton’s ring apparatus (Fig. 34–18) is immersed in a liquid, the diameter of the tenth dark ring decreases from 2.92 cm to 2.54 cm. What is the refractive index of the liquid? [Hint: See Problem 37.]

검증된 단계별 안내

Step 1: Understand the concept of Newton's rings. Newton's rings are formed due to the interference of light reflected from the top and bottom surfaces of a thin air film between a lens and a glass plate. The diameter of the rings depends on the wavelength of light and the refractive index of the medium.

Step 2: Recall the formula for the diameter of the nth dark ring in Newton's rings: \( D_n = \sqrt{4nR\lambda} \), where \( D_n \) is the diameter of the nth dark ring, \( n \) is the ring number, \( R \) is the radius of curvature of the lens, and \( \lambda \) is the wavelength of light in the medium. When the apparatus is immersed in a liquid, the wavelength changes due to the refractive index of the liquid.

Step 3: Use the relationship between the wavelength in air and the wavelength in the liquid: \( \lambda_{liquid} = \frac{\lambda_{air}}{n_{liquid}} \), where \( n_{liquid} \) is the refractive index of the liquid. This affects the diameter of the rings.

Step 4: Set up the ratio of the diameters of the tenth dark ring before and after immersion in the liquid: \( \frac{D_{air}}{D_{liquid}} = \sqrt{n_{liquid}} \). Substitute the given values for \( D_{air} = 2.92 \, \text{cm} \) and \( D_{liquid} = 2.54 \, \text{cm} \).

Step 5: Solve for \( n_{liquid} \) using the formula \( n_{liquid} = \left( \frac{D_{air}}{D_{liquid}} \right)^2 \). This will give the refractive index of the liquid.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Newton's Rings

Newton's rings are a pattern of concentric circles created by the interference of light waves reflected between a spherical surface and a flat glass plate. The dark and bright rings result from constructive and destructive interference, which depends on the wavelength of light and the radius of curvature of the lens. The diameter of these rings changes when the medium between the lens and the plate is altered, such as when immersed in a liquid.

Refractive Index

The refractive index is a dimensionless number that describes how light propagates through a medium. It is defined as the ratio of the speed of light in a vacuum to the speed of light in the medium. A higher refractive index indicates that light travels slower in that medium, which affects the interference patterns observed in experiments like Newton's rings, allowing for the calculation of the refractive index based on changes in ring diameters.

Interference of Light

Interference of light occurs when two or more light waves overlap and combine, resulting in a new wave pattern. This phenomenon can lead to regions of constructive interference (bright spots) and destructive interference (dark spots). In the context of Newton's rings, the interference pattern is influenced by the thickness of the air or liquid layer between the lens and the plate, which changes as the medium is altered, thus affecting the observed ring diameters.