Homework 22 Solutions

# Homework 22 Solutions - lim(kl9356 – Homework 22 –...

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Unformatted text preview: lim (kl9356) – Homework 22 – Weathers – (22202) 1 This print-out should have 14 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points A coin is at the bottom of a beaker. The beaker is filled with 1 . 6 cm of water ( n 1 = 1 . 33) covered by 3 . 7 cm of liquid ( n 2 = 1 . 4) floating on the water. How deep does the coin appear to be from the upper surface of the liquid (near the top of the beaker)? Correct answer: 3 . 84586 cm. Explanation: For small angles sin θ ≈ tan θ = x ℓ . The appearance of the width of the coin x remains the same. Applying Snell’s Law, n 1 sin θ 1 = n 2 sin θ 2 n i x ℓ i = n f x ℓ f ℓ f = ℓ i n i since n f ≈ 1 for air and the apparent distance d = ℓ f . Thus d = ℓ 2 n 2 + ℓ 1 n 1 = 3 . 7 cm 1 . 4 + 1 . 6 cm 1 . 33 = (2 . 64286 cm) + (1 . 20301 cm) = 3 . 84586 cm . The coin appears to be closer to the surface by (1 . 6 cm) + (3 . 7 cm) − (3 . 84586 cm) = 1 . 45414 cm . Alternate Solution: Light coming straight up from the coin falls on each in- terface at 0 ◦ and continues straight up. Con- sider light making a 1 ◦ angle (therefore we can use the small angle approximation) with the vertical in the water. It enters the liquid a distance (1 . 6 cm) tan 1 ◦ ≃ (1 . 6 cm) × θ 1 u = 0 . 0279253 cm from the vertical ray. In the liquid its angle with the vertical is given by 1 . 33 sin 1 ◦ = 1 . 4 sin θ 2 θ 2 = 0 . 950001 ◦ This same ray reaches air at distance (0 . 0279253 cm) + (3 . 7 cm) tan(0 . 950001 ◦ ) = 0 . 0892736 cm , from the vertical ray, and the angle of refrac- tion is found from 1 . 4 sin(0 . 950001 ◦ ) = 1 sin θ 3 θ 3 = 1 . 33 ◦ Your brain automatically finds the intersec- tion of this ray with the vertical ray, at an apparent depth of . 0892736 cm tan(1 . 33 ◦ ) ≃ . 0892736 cm . 0232129 rad = 3 . 84586 cm . Rays for all other small angles with the ver- tical appear to diverge from the image of the coin at this depth....
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Homework 22 Solutions - lim(kl9356 – Homework 22 –...

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