Question

雨過天晴，人們常看到天空中出現(xiàn)彩虹，它是由陽光照射到空中彌漫的水珠上時出現(xiàn)的現(xiàn)象.在說明這個現(xiàn)象時，需要分析光線射入水珠后的光路.一細束光線射入水珠，水珠可視為一個半徑為R的球，球心O到入射光線的垂直距離為d.水的折射率為n.

(1)在圖上畫出該束光線射入水珠內，經一次反射后，又從水珠中射出的光路圖；

(2)求這束光線從射向水珠到射出水珠每一次偏轉的角度.

Answer 1

(1)圖略.

(2)arcsin-arcsin 

π-2arcsin

arcsin-arcsin 

解析:

本題考查光的折射.

(1)光路如圖所示.                                                                                             (4分)

(2)以θ₁、θ₂表示入射角、折射角，由折射定律得

sinθ₁=nsinθ₂                                                                                                                                                                                                    (1分)

Δθ₁、Δθ₂、Δθ₃表示每一次偏轉的角度.由反射定律、折射定律和幾何關系

可知sinθ₁=                                                                                                (1分)

Δθ₁=θ₁-θ₂                                                                                                                                                                                                                  (1分)

Δθ₂=π-2θ₂                                                                                                                                                                                                   (1分)

Δθ₃=θ₁-θ₂                                                                                                                                                                                                                  (1分)

由以上各式解得

Δθ₁=arcsin-arcsin                                                                                (1分)

Δθ₂=π-2arcsin                                                                                        (1分)

Δθ₃=arcsin-arcsin.                                                                               (1分)