Least Time A beam of light propagates from point A in medium...

Least Time A beam of light propagates from point A in medium 1 to point B in medium $2,$ as shown in FlGURE $26-78 .$ The index of refraction is different in these two media; therefore, the light follows a refracted path that obeys Snell's law. (a) Calculate the time required for light to travel from A to B along the refracted path. (b) Compare the time found in part (a) with the time it takes for light to travel from $A$ to B along a straight-line path. (Note that the time on the straight-line path is longer than the time on the refracted path. In general, the shortest time between two points in different media is along the path given by Snell's law.)

Least Time A beam of light propagates from point A in medium 1 to point B in medium $2,$ as shown in FlGURE $26-78 .$ The index of
refraction is different in these two media; therefore, the light follows a refracted path that obeys Snell's law. (a) Calculate the time
required for light to travel from A to B along the refracted path.
(b) Compare the time found in part (a) with the time it takes for
light to travel from $A$ to B along a straight-line path. (Note that the time on the straight-line path is longer than the time on the
refracted path. In general, the shortest time between two points
in different media is along the path given by Snell's law.)

Key Concepts

Fermat's Principle of Least Time

This principle states that the path taken by light between two points is the one that requires the least time, which may not be the shortest physical distance. It underpins the behavior of light in varying media and leads to the derivation of laws such as Snell's law, showing that light will bend when encountering an interface between media with different optical properties.

Snell's Law

Snell’s law quantitatively describes how light refracts when moving between media of different indices of refraction. It relates the angles of incidence and refraction to the speeds of light in the respective media, ensuring that the path chosen minimizes travel time according to Fermat's principle.

Index of Refraction

The index of refraction is a measure of how much the speed of light is reduced when entering a medium compared to its speed in a vacuum. It plays a crucial role in determining the light’s speed and, correspondingly, the time taken to traverse a given distance, thereby affecting the path of least time when light moves through different media.

Optical Path Length

Optical path length is the product of the physical path length and the refractive index of the medium. This concept is important because it represents the effective distance light travels in a medium and is used in comparing travel times in different media, linking geometric distance with light speed variations.

Transcript

00:02 In this question, we have a beam of light propagate from point a to point b through the medium.

00:08 Then we want to show, want to compare the time taken for the light to travel from point a to point b.

00:20 In this refracted path, it will take less time than when the light takes the straight line path.

00:29 As shown.

00:32 So in part a, we want to calculate the time taken for the light to travel along the refracted path.

00:40 So to do this, again, it involves nell's law.

00:47 We are also going to do some time calculation.

00:53 So we are going to call this distance d1 and this distance d2.

01:00 And then we're also going to call this distance d2.

01:02 Angle d data 2.

01:06 And then, so we need to find data 2 first.

01:13 So using snals law, n1, side data 1 is equal to n2, side data 2.

01:26 So side data 2 is equal to, so n1 is 1 and data 1 is 45 degrees and divide by n2, which is 1 .4, and you calculate this you get 0 .505.

01:39 5 and so data 2 is equal to 30 .3 degrees.

01:48 And so we need to calculate d1 and d2.

01:54 So we have these two triangles.

02:00 This is 45 degrees.

02:03 This is d1.

02:06 And then we have another triangle.

02:10 This is d2 and then this is data 2.

02:14 So we have d1 equals 2.

02:19 So this is 50 meters, 15 meters.

02:24 Okay, so based on this information, we have d1 is equal to 50 by cosine 45 degrees, okay, and then d2 is equal to 50, divide by cosine 30 .3 degrees.

02:44 Okay, yeah.

02:46 Then you can find the time taken...

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