Concept explainers
A Fermat’s principle of least time for refraction. A ray of light traveling in a medium with speed v1 leaves point A and strikes the boundary between the incident and transmitted media a horizontal distance x from point A as shown in Figure P38.98. The refracted ray travels with speed v2 in the second medium, eventually reaching point B. The horizontal distance between points A and B is L. a. Calculate the time t required for the light to travel from A to B in terms of the parameters labeled in the figure. b. Now take the derivative of t with respect to x. What is the condition for which the ray of light will take the shortest time to travel from A to B?
Figure P38.98
(a)
The time required for the light ray to travel from the points A to B.
Answer to Problem 98PQ
The time taken for the light to travel from point
Explanation of Solution
Write the expression for the time taken for the light ray in the second medium.
Here,
Write the expression for distance travelled by the light in the second medium.
Here,
Write the expression for the time taken for the light ray in the first medium.
Here,
Write the expression for distance travelled by the light in the first medium.
Here,
Write the equation for total time taken for the light is.
Conclusion:
Substitute
Substitute
Therefore, the time taken for the light to travel from point
(b)
The required condition to travel for getting the shortest time to reach the point
Answer to Problem 98PQ
The required condition for getting the minimum time to reach the point
Explanation of Solution
Take the derivative of
From the formula
The light ray in which its shortest time taken is.
Conclusion:
Substitute
Simplify the above equation.
Therefore, the required condition for getting the minimum time to reach the point
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Chapter 38 Solutions
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