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3.14| 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 Pa = Po cos² 0 a, watts/m? 2nr2 where 0 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 Po, find the total power in watts emitted in the upper half-space by the LED; (b) Find the cone angle, 01, within which half the total power is radiated, that is, within the range 0 < 0 < 0j; (c) An optical detector, having a 1-mm² cross-sectional area, is positioned at r = 1 m and at 0 = 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 Po?

3.14| 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 Pa = Po cos² 0 a, watts/m? 2nr2 where 0 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 Po, find the total power in watts emitted in the upper half-space by the LED; (b) Find the cone angle, 01, within which half the total power is radiated, that is, within the range 0 < 0 < 0j; (c) An optical detector, having a 1-mm² cross-sectional area, is positioned at r = 1 m and at 0 = 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 Po?

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
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3.14l 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 Pa = Po cos² 0 a, 2rr2 watts/m2 where 0 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 Po, find the total power in watts emitted in the upper half-space by the LED; (b) Find the cone angle, 01, within which half the total power is radiated, that is, within the range 0 < 0 < ®;; (c) An optical detector, having a 1-mm² cross-sectional area, is positioned at r = 1 m and at 0 = 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 Po? I want the solution in handwriting, not solutions reference, in detail, especially integration, if any
Transcribed Image Text:3.14l 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 Pa = Po cos² 0 a, 2rr2 watts/m2 where 0 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 Po, find the total power in watts emitted in the upper half-space by the LED; (b) Find the cone angle, 01, within which half the total power is radiated, that is, within the range 0 < 0 < ®;; (c) An optical detector, having a 1-mm² cross-sectional area, is positioned at r = 1 m and at 0 = 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 Po? I want the solution in handwriting, not solutions reference, in detail, especially integration, if any
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