P a | 01 ed- x→ n2 02 The two points are, respectively, at perpendicular distances a and b from the interface. The displacement from P to Q has the component d parallel to the interface, and we let x represent the coordinate of the point where the ray enters the second medium. Let t = 0 be the instant the light starts from P. (a) Show that the time at which the light arrives at Q is n, Va? x2 + n2 V b2 + (d – x)² + - t = + V2 (b) To obtain the value of x for which t has its minimum value, differentiate t with respect to x and set the derivative equal to zero. Show that the result implies n2(d – x) V b? + (d – x)² ' + (c) Show that this expression in turn gives Snell's law, n, sin(@,) = n, sin(@2).

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In the figure below, a light ray travels from point P in medium 1 to point Q in medium 2. The two points are, respectively, at perpendicular distances a and b from the interface. The displacement from P to Q has the component d parallel to the interface, and we let x represent the coordinate of the point where the ray enters the second medium. Let t = 0 be the instant the light starts from P.

a) Show that the time at which the light arrives at Q is: 
b) To obtain the value of x for which t has its minimum value, differentiate t with respect to x and set the derivative equal to zero. Show that the result implies

c)Show that this expression in turn gives Snell's law,

P
a | 01
ed- x→
n2
02
The two points are, respectively, at perpendicular distances a and b from the interface. The displacement from P to Q has the component d parallel to the interface, and we
let x represent the coordinate of the point where the ray enters the second medium. Let t = 0 be the instant the light starts from P.
(a) Show that the time at which the light arrives at Q is
n, Va?
x2
+
n2 V b2 + (d – x)²
+
-
t =
+
V2
(b) To obtain the value of x for which t has its minimum value, differentiate t with respect to x and set the derivative equal to zero. Show that the result implies
n2(d – x)
V b? + (d – x)² '
+
(c) Show that this expression in turn gives Snell's law,
n, sin(@,) = n, sin(@2).
Transcribed Image Text:P a | 01 ed- x→ n2 02 The two points are, respectively, at perpendicular distances a and b from the interface. The displacement from P to Q has the component d parallel to the interface, and we let x represent the coordinate of the point where the ray enters the second medium. Let t = 0 be the instant the light starts from P. (a) Show that the time at which the light arrives at Q is n, Va? x2 + n2 V b2 + (d – x)² + - t = + V2 (b) To obtain the value of x for which t has its minimum value, differentiate t with respect to x and set the derivative equal to zero. Show that the result implies n2(d – x) V b? + (d – x)² ' + (c) Show that this expression in turn gives Snell's law, n, sin(@,) = n, sin(@2).
Expert Solution
Step 1-Ray diagram

The ray diagram is given as shownAdvanced Physics homework question answer, step 1, image 1

Step 2- Part(a)

According to the ray diagram,

In right-angle triangle PAB,

                                                        Advanced Physics homework question answer, step 2, image 1

In right-angle triangle BDQ,

                                                         Advanced Physics homework question answer, step 2, image 2

The time taken to travel from P to Q is the addition of time taken to travel from P to B and the time taken to travel B to Q.

                                                          Advanced Physics homework question answer, step 2, image 3

Here, v1 is the velocity in medium nand v2 is the velocity in medium n2.

The refractive index is given as,

                                                         Advanced Physics homework question answer, step 2, image 4

Substitute the n1 for n and v1 for v into the equation of the refractive index.

                                                          Advanced Physics homework question answer, step 2, image 5

Substitute the n2 for n and v2 for v into the equation of the refractive index.

                                                          Advanced Physics homework question answer, step 2, image 6

Substitute v1 and v2 into the expression (1).

                                                Advanced Physics homework question answer, step 2, image 7

This is the time required to reach the light at Q.

 

 

 

 

 

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