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Sterred, not shaken *March 15, 2011*

*Posted by mareserinitatis in electromagnetics, engineering, grad school, physics, science.*

Tags: fields, geometry, radians, spheres, steradian, stiridian

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Tags: fields, geometry, radians, spheres, steradian, stiridian

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A couple years ago, I was taking a class, and the professor put up a slide talking about stiridians.

What the heck is that, I thought?

It turned out that it was a misspelling. Apparently one of the references my prof chose happened to use that misspelling, and he had merely copied it. After class, I tried to politely let him know his error.

So what the heck is a *steradian*, anyway? And how would I know such an obscure word?

In order to understand what a steradian is, we should step back and look at it’s one dimensional analog, the radian. (Because I’m using an analog, I believe that qualifies me as an analog engineer!) If you’ve had trig, you’re familiar with the radian: it’s an angle in a circle that creates an arclength equal to the radius of a circle. For those of you who prefer Babylonian Units, one radian is approximately 57.3°. Graphically, it looks like this:

When you work with antennas, you generally have to work in three dimensions (unless you get lucky and have an axis of symmetry). The reason we need three dimensions is because we’re working with both electric and magnetic fields, both of which are vector quantities and change within a sphere. As an example, this shows the fields for a dipole antenna:

The electric field direction is in blue and the magnetic in red.

It turns out that when we’re describing these patterns, it’s useful to think of the surface of a sphere. We need to describe where the field is strong or weak over that sphere. Unfortunately, using a two dimensional measure of angle is inadequate for a field.

This is where the steradian comes in. If we want to describe an area of strong field, we can describe it’s span in steradians. This is a measurement of ‘unit solid angle’ – although it’s easier to think of in terms of the area of a sphere.

A steradian is the area equal to the square of the radius of the sphere. That’s it! There are 4π steradians on the surface of a sphere (similar to the 2π radians in a circle). And you can imagine that this is what it looks like:

Also, like a radian, the steradian is a unitless measure, but you can annotate it using *sr* (much like some people indicate radians with *rad*).

As I mentioned, I used this in antennas, and I actually wrote an explanation in my MS thesis. So, obscure as it may be, I had actually spent a bit of time dealing with the topic. And now that you all know about it, I sure hope that professor corrected his notes lest one of you sees it.

At least I haven’t done anything THAT bad while teaching (parroting someting to my class and not even knowing it was spelled wrong), makes me feel better.

Steradians are cool, I had an environmental optics class in grad school, who knew the world was 3-D? Too bad the prof was a little crazy and really hard to learn from.

I’ve started to think of radians as NOT being dimensionless, but rather having units of arc length or path length over units of radial length (two different kinds of length units), that way you can see how multiplying torque by angular velocity magically turns the torque newton-meters into energy newton-meters, among other things.

That antenna graphic would have been great to use in class, we just covered that stuff.

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Good point about the dimensions. I’m fortunate that most of the time, the ‘units’ (or lack thereof) get absorbed or cancelled in the formulas we’re using, but I could see the difficulty in other applications.

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…I only just now started to understand why I struggled with radians in class back in the day. Thanks, Vi Hart!

So… Steradians sound cool… but I’m not sharp enough to have all that sink in [smile].

~Luke

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I guess rather than thinking of a 1D angle, which gives you length, it’s easier to think of a 2D angle (or a radian angle in two directions, to give you an area). Unfortunately, I couldn’t find the graphic a really wanted, which shows a ‘square’ section of a sphere’s surface with a radian length on each side – so the steradian is like the area of a square, with each side having a length of one radian.

Bah. Words are hard. 🙂

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A ‘square’ with a radian length on each side has an area greater than r*r due to the bulge of said ‘square’; hence, it overestimates one steradian.

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That’s probably why I couldn’t find the graphic! 🙂 Hadn’t thought of that.

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