(II) A ray of light, after entering a light fiber, reflects at an angle of 14.5° with the long axis of the fiber, as in Fig. 32–57. Calculate the distance along the axis of the fiber that the light ray travels between successive reflections off the sides of the fiber. Assume that the fiber has an index of refraction of 1.55 and is 1.60 x 10^-4 m in diameter.

검증된 단계별 안내

Step 1: Understand the geometry of the problem. The light ray reflects off the sides of the fiber at an angle of 14.5° with the long axis of the fiber. This means the light ray forms a right triangle with the fiber's diameter as one leg and the distance traveled along the axis between reflections as the other leg.

Step 2: Identify the relationship between the angle of reflection and the geometry of the fiber. The tangent of the angle (14.5°) is equal to the ratio of the fiber's diameter (1.60 × 10⁻⁴ m) to the distance traveled along the axis between reflections. Use the formula:

\tan (θ) = \frac{d}{L}

, where θ is the angle, d is the diameter, and L is the distance along the axis.

Step 3: Rearrange the formula to solve for L, the distance along the axis:

L = \frac{d}{\tan (θ)}

. Substitute the given values: d = 1.60 × 10⁻⁴ m and θ = 14.5°.

Step 4: Calculate the tangent of 14.5° using a scientific calculator or trigonometric table. This will give you the value needed to compute L.

Step 5: Divide the fiber's diameter (1.60 × 10⁻⁴ m) by the tangent of 14.5° to find the distance L along the axis of the fiber between successive reflections. Ensure units are consistent and verify the calculation.

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주요 개념

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

Total Internal Reflection

Total internal reflection occurs when a light ray traveling in a denser medium hits a boundary with a less dense medium at an angle greater than the critical angle. This phenomenon is crucial in optical fibers, as it allows light to be trapped and guided along the fiber's length. The angle of incidence must exceed the critical angle, which depends on the refractive indices of the two media.

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. In this problem, the fiber has a refractive index of 1.55, indicating that light travels slower in the fiber than in a vacuum, which affects the angles of reflection and refraction.

Geometry of Light Path in Fibers

The geometry of the light path in optical fibers involves understanding how light travels and reflects within the cylindrical structure of the fiber. The distance traveled along the fiber's axis between reflections can be calculated using trigonometric relationships, considering the angle of reflection and the diameter of the fiber. This geometric approach is essential for determining the effective length of the light path.