Use Fermat’s principle to find the path followed by a light ray if the index of refraction is proportional to the given function

14.r-1

Short Answer

Expert verified

r=aebθ, where a=eBC1-C2and b=C1-C2, where C is a constant andB is the integration constant.

Step by step solution

01

Given Information.

The given function isr-1.Path followed by light is to be found out using Euler equations.

02

Definition of Euler equation

The Euler equations are a set of second-order ordinary differential equations that are stationary points of the given action functional in the calculus of variations and classical mechanics.

03

Use Euler equation

To find the path traversed by light in a given medium, the path taken by the light is to be minimized (time wise). Velocity of light is scaled by a factor n1in a refractive medium, then the time required to travel from point A to point B is

t=ABdt=ABvds=c1ABnds

Therefore, following integral needs to be minimized

nds=ndr2+r2θ2=n1+r2θ'2dr

Here n=r1

Therefore F=r11+r2θ'2is to be minimized

Euler equation for coordinates r,θis ddrFθ'-Fθ=0

Calculate the required derivatives

Fθ'=r2θ'r1+r2θ'2Fθ=0

Therefore,

ddrr2θ'r1+r2θ'2=0r2θ'r1+r2θ'2=Cθ'2=C2r21-C2θ'=Cr1-C2

Where Cis constant.

Integrate θ'=Cr1-C2to get the desired result

θ=Cr1-C2dr=C1-C2drr=C1-C2Inr+B

Move Bto the left side, multiply by constants to isolate rand take the exponential of the whole expression to get

eC1-C2θ-B=r

Takea=eBC1-C2and b=C1-C2to get

r=aebθ

It corresponds to two-dimensional spiral

Therefore, r=aebθ, where a=eBC1-C2and b=C1-C2, where C is a constant and Bis the integration constant.

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Most popular questions from this chapter

Find a first integral of the Euler equation for the Problem if the length of the wire is given.

Write theθLagrange equation for a particle moving in a plane ifV=V(r) (that

is, a central force). Use theθequation to show that:

(a) The angular momentum r×mvis constant.

(b) The vector r sweeps out equal areas in equal times (Kepler’s second law).

Show that the actual path is not necessarily one of minimum time. Hint: In the diagram, A is a source of light; CD is a cross section of a reflecting surface, and B is a point to which a light ray is to be reflected. APB is to be the actual path and AP'B, AP"B represent varied paths. Then show that the varied paths:

(a) Are the same length as the actual path if CD is an ellipse with A and B as foci.

(b) Are longer than the actual path if CD is a line tangent at P to the ellipse in (a).

(c) Are shorter than the actual path if CD is an arc of a curve tangent to the ellipse at P and lying inside it. Note that in this case the time is a maximum!

(d) Are longer on one side and shorter on the other if CD crosses the ellipse at P but is tangent to it (that is, CD has a point of inflection at P).

Write and solve the Euler equations to make the following integrals stationary. In solving the Euler equations, the integrals in Chapter 5, Section 1, may be useful.

4.x2x2xds

The speed of light in a medium of index of refraction n is v=dsdt=cn. Then the time of transit from AtoBis t=ABdt=c-1ABnds. By Fermat’s principle above, t is stationary. If the path consists of two straight line segments with n constant over each segment, then

ABnds=n1d1+n2d2,

and the problem can be done by ordinary calculus. Thus solve the following problems:

1. Derive the optical law of reflection. Hint: Let light go from the point A=(x1,y1)to B=(x2,y2)via an arbitrary point P=(x,0)on a mirror along thex-axis. Setdtdx=(nc)dDdx=0, where D=distanceAPB, and show that then θ=ϕ.

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