A certain light-emitting diode (LED) is centered at the origin with its surface

Question

A certain light-emitting diode (LED) is centered at the origin with its surface
in the xy plane. At far distances, the LED appears as a point, but the glowing
surface geometry produces a far-field radiation pattern that follows a raised
cosine law: that is, the optical power (flux) density in watts/m2 is given in
spherical coordinates by
Pd = P0
cos2 ?
2?r 2 ar watts/m2
where ? is the angle measured with respect to the direction that is normal to
the LED surface (in this case, the z axis), and r is the radial distance from the
origin at which the power is detected. (a) In terms of P0, find the total power
in watts emitted in the upper half-space by the LED; (b) Find the cone angle,
?1, within which half the total power is radiated, that is, within the range
0 < ? < ?1; (c) An optical detector, having a 1-mm2 cross-sectional area, is
positioned at r = 1 m and at ? = 45?, such that it faces the LED. If one
milliwatt is measured by the detector, what (to a very good estimate) is the
value of P0?

Explanation / Answer

cone angle = tan^-1(2pr 2) P0 = pdcos2q just put the values and get answer

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