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E=mc^2, the most famous equation in the world, describes the fact that anything with mass
possesses a huge amount of energy, in principle – like, a 5kg cat has enough energy in its
mass to power the entire country of Norway for a year – if only the energy could somehow
be fully extracted from the cat.
But it turns out that efficiently extracting energy from mass is a very hard thing to do.
Anti-matter is, of course, the most efficient way of extracting energy from mass since,
if you collide a cat with a cat made of anti-matter, 100% of the mass of the cat and anti-cat will
be converted into energy (powering Norway for 2 years).
But the universe has almost no naturally-occurring anti-matter, so it’s not a practical choice
for generating energy, since you’d first have to use a lot of energy to make a large
mass of antimatter.
Since we can't use antimatter, there are basically three options left to us: chemical reactions,
nuclear reactions, and gravitational reactions - aka stuff getting pulled together by gravity,
like matter falling into black holes.
Chemical reactions, for example, are so bad at extracting energy from mass that we don’t
even think about what they’re doing as converting mass to energy (even though it is).
As an illustration, reacting a balloon of hydrogen and oxygen gases creates a nice big
explosion, but the end-products of the reaction only weigh half a nanogram less than the initial
reactants , which amounts to a measly 0.00000001% efficiency of converting mass into energy.
At that rate, you’d need ten billion cats to power Norway for a year.
Nuclear reactions are a lot more efficient, but still pretty bad on an absolute scale:
splitting uranium-235 into krypton and barium converts only about 0.08% of the uranium’s
mass into energy, and fusing hydrogen into helium like in the sun converts about 0.7%
of the hydrogen’s mass into energy.
At that rate you’d need 150 cats to power Norway for a year.
This where black holes come in – they’re about as good as it gets in our universe for
extracting energy from mass.
Which may sound weird, because, as you’ve probably heard, nothing can escape black holes
– once inside.
But the efficiency of black holes comes from what stuff does while falling towards them,
before passing the no-turning-back point of the event horizon.
Anything that falls in a gravitational field speeds up, gaining kinetic energy, and if
it then crashes into something it can convert that kinetic energy into heat.
That heat can then radiate away as infrared radiation, slightly decreasing the mass of
the object.
For planets and stars, this conversion of mass into energy is pretty pathetic: an object
falling to the surface of the earth releases only about one billionth of its mass as energy.
That’s basically as bad as a chemical reaction!
But black holes have something special going for them: they are stupendouslysmall.
A black hole with the mass of the earth would be about 2 cm across, providing way farther
for an object to fall – and since gravity gets stronger and stronger the closer you
are to an object, objects falling into black holes get accelerated to ridiculous speeds.
Specifically, an object falling all the way to the event horizon of a black hole will
have kinetic energy equivalent to converting roughly half of its half of its E=mc2 mass
energy mass.
However, if the object continues to fall into the black hole, all of that energy will be
stuck inside the black hole.
The way to actually convert mass into energy that goes out into the universe is to have
the object slowly spiral into the black hole, crashing into other stuff, heating up, radiating
that energy away thereby losing mass and speed, slowing down more, spiraling to a yet lower
orbit, and so on, all the way down to the innermost possible orbit.
And this is exactly what accretion disks around black holes do!
So how good are they at converting mass to energy?
Well, for a non-rotating black hole, the innermost possible circular orbit is actually 3 times
farther out than the event horizon, and in order to spiral in to that point an object
has to convert around 6% of its mass into energy radiated away to the outside universe.
After that point if it loses any more energy it’ll plunge down into the black hole, after
which no more energy can be extracted.
But at this 6% rate, you’d only need to throw 17 cats into a black hole to power Norway
for a year.
Compared to the 0.00000001% efficiency of chemical reactions and the 0.7% efficiency
of nuclear reactions, 6% for a non-rotating black hole may seem pretty good.
But rotating black holes are even better, because of how they bend spacetime.
They literally “drag” things orbiting them in the direction of their rotation, which
means the innermost possible orbit can be much closer to the black hole (as long as
you’re rotating along with the black hole).
The details depend on how fast the black hole is rotating, but for a very quickly rotating
black hole the innermost possible orbit coincides with the event horizon!
And the event horizon itself is half as big as for a non-rotating black hole.
Combined together, this means that matter falling into rotating black holes can convert
as much as 42% of its mass into energy.
Or equivalently, you’d only need 2 and a half inspiralling cats to power Norway for
a year.
So, if you really want to convert the mass of an object into energy, don’t bother with
chemical reactions, or nuclear fission, or nuclear fusion: throw it into a rapidly rotating
black hole.
If you’re wondering how I calculated the efficiencies of converting mass to energy,
you can just divide the energy any reaction releases by the mass energy of the things
involved – for example, when radium radioactively decays into radon and helium it releases 6.6
MeV of energy, and the mass energy of a single neutron or proton is about 940MeV, so I’ll
leave it to you to figure out how efficient alpha decay is at converting mass to energy!
Or you can learn more about nuclear fission and fusion by finishing this quiz on Brilliant.org,
which is this video’s sponsor and is full of interactive quizzes and mini courses on
physics and math.
If you really want to understand physics deeply, you have to work through calculations and
solve problems yourself, and Brilliant offers an interactive online way to do just that.
You can check out their course on black holes for free using the link in the description,
and if you decide to sign up for premium access to all of their courses, you can get 20% off
by going to Brilliant.org/minutephysics.
Again, that’s Brilliant.org/minutephysics which lets Brilliant know you came from here.


The Unreasonable Efficiency of Black Holes

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林宜悉 發佈於 2020 年 3 月 28 日
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