Read the Beforeitsnews.com story here. Advertise at Before It's News here.
Profile image
By An Objectivist Individualist (Reporter)
Contributor profile | More stories
Story Views
Now:
Last hour:
Last 24 hours:
Total:

Calculating the Cross Section of Planar Radiation Upon a Sphere and Finding the Average Radiation Intensity Over the Sphere

% of readers think this story is Fact. Add your two cents.


The incidence of planar radiation upon a sphere, such as that from the sun upon the Earth, is an important calculation for climate science, so I am going to demonstrate that calculation.  Some people have doubts about the common claim that the average incident intensity of solar insolation upon the Earth per unit area is one-quarter of the radiation intensity at that part of the Earth where the surface is at normal incidence to the solar radiation.

Let the amplitude of the planar radiation be P0, which is commonly expressed as W/m2.   The incident radiation on the sphere surface normal to the planar radiation will have an intensity of P0.  As one examines a unit surface are of the sphere which is non-normal to the planar wave of radiation, the unit area of the sphere intercepts less and less of the planar radiation the greater the surface normal angle is with respect to the direction of the plane wave.  Let us call the angle from the center of the sphere to the point on the surface upon which the planar wave has normal incidence and the surface annulus of points equidistance from the normal incidence point θ.  The surface of the hemisphere facing the planar radiation then consists of a normal point and a series of annuli, one for each Δθ that we choose.  The annuli have increasing radii perpendicular to the planar wave of radiation.  The angle θ to describe these annuli varies from 0 to π/2 radians or 90̊.  We will use the radian units.  Let the radius of the sphere be R.

Each annulus has a circumference of 2πR sin θ and it intercepts an annulus of the planar radiation wave with the same circumference, but with a narrower width than the annulus on the sphere surface has for θ > 0.  The width of the planar wave annulus that projects onto the sphere surface is given by Δθ R cos θ.  Thus, at angle θ the area of the plane wave of radiation intercepted by the sphere at angle θ, is

AI(θ) = 2πR2 sin θ cos θ Δθ

Integrating over the range from 0 to π/2 for θ, we find the area of the planar wave of radiation which is intercepted by the sphere.  The integral is

AI= 2πR20π/2 sin θ cos θ dθ

The value of the integral is ½ sin2θ, which for these limits of integration is ½.  Consequently,

AI= πR2

The total incident planar wave intensity on the sphere (all on one hemisphere) is then

PT= P0 AI = πR2 P0

The total area of the sphere, AS, is 4πR2, so we can rewrite this as

PT= (1/4) ASP0

Averaging over the entire area of the sphere, the total incident plane wave intensity average per unit area is then P0/4.



Source: https://objectivistindividualist.blogspot.com/2019/09/calculating-cross-section-of-planar.html



Before It’s News® is a community of individuals who report on what’s going on around them, from all around the world.

Anyone can join.
Anyone can contribute.
Anyone can become informed about their world.

"United We Stand" Click Here To Create Your Personal Citizen Journalist Account Today, Be Sure To Invite Your Friends.

Please Help Support BeforeitsNews by trying our Natural Health Products below!


Order by Phone at 888-809-8385 or online at https://mitocopper.com M - F 9am to 5pm EST

Order by Phone at 888-388-7003 or online at https://www.herbanomic.com M - F 9am to 5pm EST

Order by Phone at 888-388-7003 or online at https://www.herbanomics.com M - F 9am to 5pm EST


Humic & Fulvic Trace Minerals Complex - Nature's most important supplement! Vivid Dreams again!

HNEX HydroNano EXtracellular Water - Improve immune system health and reduce inflammation

Ultimate Clinical Potency Curcumin - Natural pain relief, reduce inflammation and so much more.

MitoCopper - Bioavailable Copper destroys pathogens and gives you more energy. (See Blood Video)
Oxy Powder - Natural Colon Cleanser!  Cleans out toxic buildup with oxygen! 
Nascent Iodine - Promotes detoxification, mental focus and thyroid health.
Smart Meter Cover -  Reduces Smart Meter radiation by 96%!  (See Video)

Immusist Beverage Concentrate - Proprietary blend, formulated to reduce inflammation while hydrating and oxygenating the cells.

Report abuse

    Comments

    Your Comments
    Question   Razz  Sad   Evil  Exclaim  Smile  Redface  Biggrin  Surprised  Eek   Confused   Cool  LOL   Mad   Twisted  Rolleyes   Wink  Idea  Arrow  Neutral  Cry   Mr. Green

    MOST RECENT
    Load more ...

    SignUp

    Login

    Newsletter

    Email this story
    Email this story

    If you really want to ban this commenter, please write down the reason:

    If you really want to disable all recommended stories, click on OK button. After that, you will be redirect to your options page.