APPENDIX 1
SOLID ANGLE 
2D angle
Let us remind ourselves the definition of a two-dimension (2D) angle  (in radians).
Fig. 1 (a) shows this angle subtended by an arc of length l of a circle of radius r.
fig. 1 (a) Showing an angle  (in radians) subtended by an arc length l and over a
distance r.
The angle is defined as
 = l/r
(1)
Thus around the whole circle of 360 degrees, the angle in radians is
 = 2r/r = 2
(2)
Using ratio, the conversion from an angle of x degrees to  radians is
 = 2×x/360 = ×x/180
(3)
If x = 90 degrees, we have
 = /2 radians
(4)
3D angle (solid angle)
For a solid angle normally denoted by  of d, the situation is similar, see fig. 1 (b).
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Fig. 1 (b), Showing a solid angle d (in radians) subtended by an area A at the origin
over a distance r.
The arc of a circle is now replaced by an area element A on a sphere of radius r. The
shape of the area A can be any irregular shape as long as it is a part of the surface of the
sphere. The 3D angle  subtended by A at the centre (origin) of the sphere is defined as the
solid angle. This is defined mathematically as:
 = A/r2
(5)
Remembering that A is measured in area with a dimension of l2, (5) means that the
solid angle  here has no dimension, i.e. a pure number (in radians).
For the application of kinetic theory (see appendix 2), we use a small solid angle d
subtended by the element A. A is defined by the polar coordinates elements on the surface of
the sphere. These elements are:
rd along the angle  measured from the z-axis and
rsind along the angle  measured from the x-axis
(6)
With these 2 elements, the area is given by
A = rd×rsind = r2×sindd
(7)
From (5), the solid angle element is now given by
d = A/r2 = sindd
(8)
Eq (8) is very useful in kinetic theory as can be seen in appendix 2.
Let us now test the formula and find the total solid angle  for the whole sphere. This
is equivalent to the whole circle in the 2D case which has a total angle of 2 eq(2).
 = d = 0sind×02d = 2×2 = 4
(9)
Thus the solid angle for the whole sphere is 4.
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APPENDIX 2
NUMBER OF PARTICLES STRIKING UNIT AREA PER SECOND (¼nv)
Consider a cylinder of length v in the direction shown in fig. 2. To set up our integral,
we only include molecules travelling within the small solid angle of d as shown. We now
consider a group of molecules with speed v to v+dv in this direction.
Fig. 2, Showing a thin cylinder whose axis is along the direction of the velocity v.
The intercept of the cylinder on the x-y plane is the aperture of area D and the small solid
angle d defines the small range of speed directions under consideration.
If the area shown in the x-y plane at the origin O is D, the cross-sectional area of the
cylinder A is
A = D×cos
(10)
The number density with speed v to v+dv is
dn = n(v)dv
(11)
This is a scalar quantity basically including v of all direction of solid angle 4.
However, we only want to consider a much smaller group of molecules travelling along the
direction of the cylinder and also within a small spread of solid angle d. Therefore, we need
to modify dn to this special group of molecules in d.
dn’ = dn×(d/4)
(12)
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The general technique to calculate the number of particles crossing normally an area
A is to use the argument of emptying a cylinder in one second, i.e. length of cylinder being v.
The no. of such molecules crossing the cross-section (of area A) of the cylinder per unit time
is given by
dn’/dt = (cross-sectional area)×(length of cylinder)×concentration
= A×v×dn’ = Dcos×v×dn×(d/4)
= Dsincosddvn(v)dv/4
(13)
To find the total rate for all speeds v and all the solid angle above the x-y plane, we
integrate as follows:
dn/dt = D0½sincosd×02d×0v.n(v)dv/4
= D½×2×nv/4
= nvD/4
(14)
where v is the average speed and n is the total concentration of the molecules.
The definition of average speed v defined by
0v.n(v)dv = nv
(15)
is from eq (36) of my application notes.
To find the no. striking per unit area per second, we divide (14) throughout by the aperture
area D, thus
dn/dt = ¼nv
(16)
Last updated:
22 January 2015
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