Physics for Scientists and Engineers with Modern Physics, Technology Update
Physics for Scientists and Engineers with Modern Physics, Technology Update
9th Edition
ISBN: 9781305401969
Author: SERWAY, Raymond A.; Jewett, John W.
Publisher: Cengage Learning
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Chapter 35, Problem 84CP

(a)

To determine

The time at which the light arrives at Q is given by t=r1v1+r2v2=n1a2+x2c+n2b2+(dx)2c.

(a)

Expert Solution
Check Mark

Answer to Problem 84CP

The time at which the light arrives at Q is given by t=r1v1+r2v2=n1a2+x2c+n2b2+(dx)2c.

Explanation of Solution

Write the expression to obtain the speed of light in medium 1.

    v1=cn1

Here, v1 is the speed of light in medium 1, c is the speed of light in air and n1 is the refractive index of the medium 1.

Write the expression to obtain the speed of light in medium 2.

    v2=cn2

Here, v2 is the speed of light in medium 2, c is the speed of light in air and n2 is the refractive index of the medium 2.

Write the expression to obtain the time when light arrive at point Q.

    t=r1v1+r2v2

Here, t is the time when light arrive at point Q, r1 is the distance travelled by the light in medium n1, v1 is the velocity of light in medium n1, r2 is the distance travelled by the light in medium n2 and v2 is the velocity of light in medium n2.

Substitute cn1 for v1 and cn2 for v2 in the above equation.

    t=r1(cn1)+r2(cn2)=r1n1c+r2n2c                                                                                                     (I)

The light ray path in two mediums is as shown in the figure below.

Physics for Scientists and Engineers with Modern Physics, Technology Update, Chapter 35, Problem 84CP

Figure-(1)

Write the expression to obtain the value of r1 based on Pythagoras theorem in triangle PMN.

    r1=a2+x2

Here, r1 is the distance travelled by the light in medium n1.

Write the expression to obtain the value of r2 based on Pythagoras theorem in triangle QNO.

    r2=(dx)2+b2

Here, r2 is the distance travelled by the light in medium n2.

Substitute a2+x2 for r1 and (dx)2+b2 for r2 in equation (I).

    t=(a2+x2)n1c+((dx)2+b2)n2c                                                               (II)

Therefore, the time at which the light arrives at Q is given by t=r1v1+r2v2=n1a2+x2c+n2b2+(dx)2c.

(b)

To determine

The result is n1xa2+x2=n2(dx)b2+(dx)2 when the value of t is differentiated with respect to x and equate it to zero.

(b)

Expert Solution
Check Mark

Answer to Problem 84CP

The result is n1xa2+x2=n2(dx)b2+(dx)2 when the value of t is differentiated with respect to x and equate it to zero.

Explanation of Solution

Consider the equation (II).

    t=(a2+x2)n1c+((dx)2+b2)n2c

Differentiate the above equation with respect to x.

    dtdx=n1c(ddx(a2+x2))+n2c(ddx((dx)2+b2))=n1c(12a2+x2(2x))+n2c(12(dx)2+b2)2(dx)(1)=n1xca2+x2n2(dx)cb2+(dx)2

Substitute 0 for dtdx in the above equation.

    0=n1xca2+x2n2(dx)cb2+(dx)2n1xca2+x2=n2(dx)cb2+(dx)2n1xa2+x2=n2(dx)b2+(dx)2                                                        (III)

Therefore, the result is n1xa2+x2=n2(dx)b2+(dx)2 when the value of t is differentiated with respect to x and equate it to zero.

(c)

To determine

The expression of Snell’s law is given as n1sinθ1=n2sinθ2.

(c)

Expert Solution
Check Mark

Answer to Problem 84CP

The expression of Snell’s law is given as n1sinθ1=n2sinθ2.

Explanation of Solution

Write the expression to obtain the value of sinθ1 in triangle PMN.

    sinθ1=xx2+a2

Write the expression to obtain the value of sinθ2 in triangle QNO.

    sinθ2=dxb2+(dx)2

Substitute xx2+a2 for sinθ1 and dxb2+(dx)2 for sinθ2 in equation (III).

    n1sinθ1=n2sinθ2

Therefore, the expression of Snell’s law is given as n1sinθ1=n2sinθ2.

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Chapter 35 Solutions

Physics for Scientists and Engineers with Modern Physics, Technology Update

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