Refraction and Lenses Legacy Problem #13 Guided Solution
Problem*
Determine the critical angle of the following materials when surrounded by air:
- Teflon (n = 1.38)
- Pyrex glass (n = 1.47)
- Polycarbonate glass (n = 1.59)
- Sapphire gemstone (n = 1.77)
- Diamond (n = 2.42)
Audio Guided Solution
The critical angle is the angle of incidence which causes the angle of refraction to be 90 degrees. When light approaches a boundary at the critical angle, then the refracted ray will lie along the boundary between the more dense and the less dense material. Critical angles are only experienced when light is traveling through the more dense material, like Teflon, and approaching the less dense material, like air. In order to calculate the critical angle, we need to use Snell's Law. In Snell's Law, we need to plug values of the indices of refraction of the two materials with the index of refraction of the critical angle material being associated with the unknown quantity, theta i. So in the first case here, we'll go 1.38 times the sine of theta i, where theta i is the critical angle. And that's equal to 1.00 for air times the sine of 90 degrees. Now we need to solve this problem for the critical angle. The right side of the equation evaluates to 1. The left side of the equation has the 1.38 times the sine of the unknown. So if we divide each side of the equation by 1.38, we end up with the sine of the critical angle equal 1 divided by 1.38. Now if we take the inverse sine of each side of the equation, we'll have the critical angle. That's the second sine button on most calculators. The second sine of 1 divided by 1.38. That will give us a critical angle value for the Teflon air boundary of about 46.4 degrees. We can repeat the process for the other parts of this question, simply replacing the 1.38 with the 1.47.
Solution
- 46.4°
- 42.9°
- 39.0°
- 34.4°
- 24.4°
Habbits of an Effective Problem Solver
- Read the problem carefully and develop a mental picture of the physical situation. If necessary, sketch a simple diagram of the physical situation to help you visualize it.
- Identify the known and unknown quantities and record them in an organized manner. Equate given values to the symbols used to represent the corresponding quantity - e.g., \(\descriptive{d_o}{d_o,distance object} = 24.8\unit{cm}\); \(\descriptive{d_i}{d_i,distance image} = 16.7\unit{cm}\); \(\descriptive{f}{f,focal length} = \colorbox{gray}{Unknown}\).
- Use physics formulas and conceptual reasoning to plot a strategy for solving for the unknown quantity.
- Identify the appropriate formula(s) to use.
- Perform substitutions and algebraic manipulations in order to solve for the unknown quantity.
Read About It!
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