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Find the solid angle subtended at the orgin

  1. Feb 15, 2004 #1
    In the following two questions, how would I find the solid angle at the origin?

    1.Find the solid angle subtended at the orgin by the triangle with corners P=(1,0,0) Q=(0,1,0) and R=(0,0,1).

    2.What is the solid angle subtented at the orgin by the plane x+y=4?


    For the second question, will i start by making x=0 and solve for y. So when x=0 y=4. y=0 x=4 (0,4) (4,0) Then what?
     
  2. jcsd
  3. Feb 16, 2004 #2

    HallsofIvy

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    I think what you need is this: the measure of the central angle of an entire circle, in radians, is 2π precisely because a circle of radius 1 has circumference of 2π. Of course, since a right angle cuts exactly 1/4 of a circle, the measure of a right angle in radians is (2π)/4= π/2.

    You didn't say what units you want the measure of the angle in but I will assume you want it in "steradians" which are defined in a way similar to radians: The surface area of a sphere with radius 1 is 4π so the "solid-angle" of an entire sphere is 4π steradians.

    The three points you give, P=(1,0,0) Q=(0,1,0) and R=(0,0,1), are on the positive x, y, z axes so the solid angle you are talking about is precisely the "first octant"- one eighth of an entire sphere. In steradians, the measure is (4π)/8= π/2.


    I honestly don't know what you mean by "the solid angle subtended at the orgin by the plane x+y=4" since the entire plane does not have an edge. To draw all possible lines from from the origin to that plane you would have lines arbitrarily close to the plane x+y= 0. The best answer I could give to the question as you phrased it would be to say that the plane subtends an entire "hemisphere" and so would have measure 2π.

    But you talk about setting x= 0 and calculating y. You seem to be talking about the portion of the plane x+y= 0 lying in one quadrant.
    Of course, when x= 0, y= 4 and when y= 0, x= 4. But there is no "z" in the equation. x+ y= 4 is a plane parallel to the z-axis. If you really mean to cut it off that way, you need to include both the first and eight quadrants (z takes on all possible values both positive and negative) and so the measure would be 4π/2= 2π steradians.

    If you had the plane x+y+z= 4, cutting the axes at (4,0,0), (0,4,0), (0,0,4), the solid angle subtended by the plane in the first quadrant would still be the entire first quadrant: measure
    π/2 steradians as in the first problem. The distances from the origin to the points where the plane cuts the axes are not relevant- just as the lengths of two sides of a triangle are irrelevant to the measure of the angle between them.
     
    Last edited: Feb 16, 2004
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