Exactly. People build telescopes (mirror or lenses) with longer focal length to get bigger images so they can see more detail in the celestial objects. Then they have to make the diameter bigger to capture enough light to see that detail without a very long exposure.
Do this thought experiment.
Draw a point A which epresents one spot on your magic lense or mirror. From A, draw a line to the center of the sun S. then draw 2 lines to the edge of the sun’s discS1 and S2. (The disc is ½° so there will be one line with two others, one on each side spreading ¼° from that centerline AS.
Those lines reflect or refract at an angle to target T 100’ away. Since the mirror or lens does what they are supposed to do, each line changes direction the same amount.
So now draw lines AT to target, and the lines from the edge of the suns disc will also be ¼° from AT at AT1 and AT2
The same applies to every spot on the lens (or mirror) they all aim at the same point T.
100 feet at ¼° is 0.44 feet or 5.24" so draw a circle radius 5.24" (diameter 10.5") around T and that is the smallest image of the sun you can get with a 100 foot focal length mirror or lens.
When you are frying ants or burning paper, you are taking a 3" or so glass and concentrating all that solar energy into a spot a tiny fraction of an inch diameter.
Let’s say your paper fire starter lens is focal length 6" so produces about 1/20" spot. Ratio of 3:0.05 or 60:1 concentration.
For the same effect at 100 feet you would need a lense or mirror 60x10.5" = 630" or 52’ 6" diameter.
Actually, the reason that early telescopes had long focal lengths was because they hadn’t yet learned enough about figuring a mirror to properly parabolize it and get rid of spherical aberration. The longer the focal length, the closer the shape isd to a parabola and the smaller the error. They also had problems with chromatic aberration that were minimized with lng focal length mirrors:
One of my uni professors had a small amount of NASA funding to investigate vacuum pump backed mirrors for space telescopes. They would have allowed changes to the focal length and corrections to any manufacturing defects. He got a lot of schadenfreude from his work not being included on Hubble.
That might have been the dominant reason for early telescopes, but even modern astronomical telescopes usually have fairly long focal lengths, because resolution is usually something that’s desired.
I hope I didn’t come across too harshly, I agree that the Mythbusters were too hasty in their dismissal.
The fact that someone was able to set tarred hemp on fire made me think of one use of this method. Simply send out a highly combustible raft and set it ablaze from a distance and inspire awed terror.
On the other hand, I have to imagine that even back then everyone was familiar with the concept of reflecting sunlight off a shiny surface so maybe terror but not awe. But if you were in range of that weapon you may soon be in range of the Claw of Archimedes which I gotta say scares me 2000 years later. Clever chap, that Archimedes.
But resolution doesn’t require long focal length. Resolution requires a large aperture. The Hubble and other space telescopes are necessarily a lot smaller than those early scaffold-mounted telescopes partly because they have to be launched in a vehicle, but if they had to be long they could be assembled in orbit (with, I admit, quite a bit of difficulty). They haven’t been, because it’s not needed. But all of them have large apertures.
Of course, it becomes difficult to have low aberration with a short focal length yet large aperture telescope, but it’s doable.
Focal length still helps. If nothing else, eventually you run up against the physical size of the pixels (or equivalent) on your sensor. It does you no good to have an optically-sharp image at your sensor, if that image is itself too small to fully resolve.
Granted, we can make some mighty small sensors these days (driven largely by the phone market), but this would have been an issue for, for instance, the Hubble.
Spy satellites use folded optics to achieve very long focal lengths. This is practical because they’re imaging the Earth’s brightly sunlit surface as opposed to a dim astronomical object. As a result they can blow up an extraordinarily tiny angular view to the limits of the resolution of the primary.
What a gift you are.