Chapter 9: Q14P (page 482)
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.
Short Answer
, where and , where is a constant and is the integration constant.
Chapter 9: Q14P (page 482)
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.
, where and , where is a constant and is the integration constant.
All the tools & learning materials you need for study success - in one app.
Get started for freeFind 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.
The speed of light in a medium of index of refraction n is . Then the time of transit from is . By Fermat’s principle above, t is stationary. If the path consists of two straight line segments with n constant over each segment, then
,
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 to via an arbitrary point on a mirror along the. Set, where , and show that then .
What do you think about this solution?
We value your feedback to improve our textbook solutions.