Thread: Silly question (probably any HS student could figure) inspired by some other post

Care of Doctor Rick

We can't say how many degrees there are in a sphere, any more than we
can say how many feet there are in an acre. Feet are a measure of
length, and an acre is an area, not a length. You can't measure an
area with a tape measure. Likewise, degrees are a measure of an angle;
you can sweep out a circle by swinging a line through an angle of 360
degrees. But you can't sweep out a sphere by swinging a line through
some angle, so angle measure won't do to measure a sphere.

Let's think more about the analogy to length versus area. We can
measure area in *square* feet. Is there anything like "square degrees"
that we can use to measure a sphere? Yes, there is! But instead of
degrees, we start with radians, a different measure of angles. We come
up with something that could perhaps be called "square radians."
Squares won't really enter into it, though, so instead we call the
unit a "steradian" (like "stereo radian"; stereo is from the Greek for
solid, or 3-dimensional). We say that it is a measure of "solid
angle."

Do you know the idea behind the radian measure of an angle? You draw a
circle using the vertex of the angle as center. Then measure the
length of the arc cut off by the two legs of the angle, and divide
this length by the radius of the circle. The ratio is the same no
matter what size circle you draw; we call the ratio the radian measure
of the angle.

If you do this with a full circle (a 360-degree angle), then the arc
is the full circumference of the circle. Its length is 2 pi times the
radius of the circle. Divide this by the radius, and you get 2 pi.
Thus 360 degrees equal 2 pi (approximately 6.28) radians.

What is a solid angle? One way to picture a solid angle is the tip of
a cone or a pyramid. A tall narrow cone has a small solid angle at the
tip; a broad flat cone has a large solid angle at the tip. The solid
angle doesn't have to be "round" though. Just as you can have
different shapes with the same area, you can have solid angles with
different "shapes" but the same measure (in steradians). For instance,
the peak of a triangular pyramid is sort of a "triangular" solid
angle, and the peak of a square pyramid is sort of a "square" solid
angle.

I have shown you how the measure of an angle is related to the length
of an arc. Now let's think about a sphere and a "solid angle." Take
the peak of that cone or pyramid, and draw a sphere around it. (You'll
have to imagine this; I can't draw in the air.) The solid angle cuts
off a piece of the sphere. If we measure the area of this piece, and
divide the area by the square of the radius of the sphere, then we
have a measure of the solid angle in steradians.

The surface area of a sphere is 4 pi times the square of the radius.
Therefore the entire sphere has a solid angle of 4 pi steradians.
That's as close as we're going to get to an answer for your question:
how many degrees are there in a sphere?

Thanks. That clears it up nicely, and as well, adds another obscure unit of measurement to the way I take in the physical universe...

360*180=64800 possibilities I am thinking.

Originally Posted by sfay93

360*180=64800 possibilities I am thinking.

if you are only dividing it into 1 degree units.

And the wireless power transmission has been done many times on small scale. Think those pads that you just rest your cell phone or camera on and the batteries charge. It is directional sort of. More "up" from the flat pas than speherical radiation, but there is still a semi-sphere that occurs but the farther you go, the less powerful the wave.

They are actually developing this technology now to carry vast amounts of energy a meter or two and have been relatively successful so far. Think no more wall outlets. Just as long as your device is within a meter or two from your house, it is powered as if connected to the mains by a cord. The floor and ceiling would be perfect for placing the transmitter because you could move it anywhere.

Pretty nifty stuff, and not that far away (10yrs maybe).

Would be cool. No powerwire running through the car anymore

On a simpler level, some guy in town decided to steal power from the utility company wirelessly, and he gets away with it because there are no power lines runnning into his house!

Basically in his yard he put up HUGE wire coils. And he lives across the street from a power distribution hub thingy (whatever those are... Stepdown transformers from the ultra-high power to semi-high power into the little boxes that make it only 220). He was using the huge coils to trap the power and ran his entire house off of it! Crazy!

Yay to free electricity

Yeah, 138KV lines put out some serious Magnetic Flux. I wonder just how big his coils were. I could do some theory to practice, since that's what I do for a living. I work at a small, 90MW, cogeneration power plant. But I'd have to put the doughnut down, and actually get up...doh.

The coils were almost a story high! I said HUGE!