Find the solid angle subtended at the orgin

[jlmac2001]

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?

 

[HallsofIvy]

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.

 

[M98Ranger]

To find the solid angle subtended at the origin, you can use the formula for solid angle, which is given by Ω = A/R^2, where A is the area of the surface and R is the distance from the origin to the surface.

1. For the first question, you can find the area of the triangle by using the formula for the area of a triangle, which is given by A = 1/2 * base * height. In this case, the base would be PQ and the height would be the distance from the origin to the plane containing the triangle, which is equal to 1/√3. So the area of the triangle would be A = 1/2 * √2 * 1/√3 = √2/6.

Next, you need to find the distance from the origin to the triangle. This can be done by finding the length of the vector OP, where O is the origin and P is one of the corners of the triangle. In this case, OP = √(1^2 + 1^2 + 1^2) = √3.

Now, you can plug in the values into the formula for solid angle to get Ω = (√2/6)/√3^2 = √2/18.

2. For the second question, you can find the area of the plane by solving for y in terms of x in the equation x + y = 4. This gives you y = 4 - x. Now, you can use the formula for the area of a triangle, where the base is the distance between the points (0,4) and (4,0) and the height is the distance from the origin to the plane, which is equal to 4/√2. So the area of the plane would be A = 1/2 * 4 * 4/√2 = 4√2.

Next, you need to find the distance from the origin to the plane. This can be done by finding the length of the normal vector to the plane, which is (1,1,0). So the distance would be R = √(1^2 + 1^2 + 0^2) = √2.

Now, you can plug in the values into the formula for solid angle to get Ω = (4