Pierre de Fermat (1601–1665) showed that whenever light travels from one point to another, its actual path is the path that requires the smallest time interval. This statement is known as Fermat's principle. The simplest example is for light propagating in a homogeneous medium. It moves in a straight line because a straight line is the shortest distance between two points. Derive Snell's law of refraction from Fermat's principle. Proceed as follows. In Figure P34.54, a light ray travels from point Pin medium 1 to point Q in medium 2. The two points are, respectively, at perpendicu- lar 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 Qis a² + x² n2 Vb? + (d – x)² t= -+ (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 n,(d – x) Va? + V6?+ (d – x)² (c) Show that this expression in turn gives Snell's law, п, sin 6, 3 п, sin 0, d PN a i 0, ed - x+ 02 Figure P34.54 Problems 54 and 55.
Pierre de Fermat (1601–1665) showed that whenever light travels from one point to another, its actual path is the path that requires the smallest time interval. This statement is known as Fermat's principle. The simplest example is for light propagating in a homogeneous medium. It moves in a straight line because a straight line is the shortest distance between two points. Derive Snell's law of refraction from Fermat's principle. Proceed as follows. In Figure P34.54, a light ray travels from point Pin medium 1 to point Q in medium 2. The two points are, respectively, at perpendicu- lar 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 Qis a² + x² n2 Vb? + (d – x)² t= -+ (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 n,(d – x) Va? + V6?+ (d – x)² (c) Show that this expression in turn gives Snell's law, п, sin 6, 3 п, sin 0, d PN a i 0, ed - x+ 02 Figure P34.54 Problems 54 and 55.
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Transcribed Image Text:Pierre de Fermat (1601–1665) showed that whenever light
travels from one point to another, its actual path is the path
that requires the smallest time interval. This statement
is known as Fermat's principle. The simplest example is for
light propagating in a homogeneous medium. It moves in a
straight line because a straight line is the shortest distance
between two points. Derive Snell's law of refraction from
Fermat's principle. Proceed as follows. In Figure P34.54,
a light ray travels from point Pin medium 1 to point Q in
medium 2. The two points are, respectively, at perpendicu-
lar 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 Qis
a² + x²
n2 Vb? + (d – x)²
t= -+
(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
n,(d – x)
Va? + V6?+ (d – x)²
(c) Show that this expression in turn gives Snell's law,
п, sin 6, 3 п, sin 0,
Transcribed Image Text:d
PN
a i 0,
ed - x+
02
Figure P34.54 Problems 54 and 55.
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