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TTT's first question...
First let's set the scene. It's the early 1900s, and JJ Thomson recently discovered the electron (the first subatomic particle to be identified), while Einstein and others have finally convinced the world that atoms are real. So the race is on to discover their structure. At the same time, Marie Curie has investigated radioactivity and people have learned how to harness it so that they can create alpha particles at will.
From the point of view of early 20th-Century physics, an alpha particle is a tiny, positively-charged, super-fast, high-energy bullet. It doesn't travel very far before hitting something, but the beauty part of that is that you know where it hits and you know where it's heading**. The other point is that, by this time already, it's well-known that if two things hit each other then you can use well-founded principles like "conservation of momentum" and "conservation of energy" to deduce something about the collision. So this leads to an idea that is the basis of all modern particle physics experiments: fire two things together and see what happens when they smash into each other.
The Geiger-Marsden-Rutherford experiment is one of the first of these. They hit upon the idea to try and work out what atoms look like by firing alpha particles at an atom and see what happens. By this point, as I say, it was already known that atoms have electrons inside them, but also that atoms are electrically neutral, so there must be a positive charge in the atom *somewhere*. But where, and how is it spread out?
The most reasonable first idea is that, because an atom is a hard ball and because the electrons are tiny, then they might swim around in a "sea" of smeared-out positive charge. On this scale, the positive charge would be so smeared out that any speeding bullet should burst through it, perhaps with a small deflection/glancing blow on the way. If that's the case, then you should see alpha particles on the other side of any (thin) target.
And, mostly, this is what was observed. So far, so good. But... luckily, the scientists involved thought to set up a detector capable of seeing if any alpha bullets were deflected by more than just a slight angle. And, astonishingly, some were! Not many, but also far more than anybody could expect. The classic quote is along the lines of firing a cannonball at tissue paper and watching it bounce back into your lap.
The two key observations, then, are:
1. Some alpha particles fired at a thin target (gold foil) deflect almost all the way back on themselves;
2. Most alpha particle pass straight through the foil.
The way Rutherford reconciled this was to conclude that perhaps the positive charge in the atom was concentrated into a tiny spot! This works, because:
1. Positive charges repel, so if the alpha particle heads directly towards the concentrated charge then it should be repelled backwards;
2. Most of the atom is empty space, allowing the alpha particle to pass straight through unimpeded.
The particular genius of Rutherford is not just to describe this model but also to use it to make a testable prediction. Actually, his second genius was to publish this model knowing that, according to the then known laws of physics, it made no sense at all.*** He pressed ahead because it fit with experimental data (a key distinction, for anybody lurking -- the hold standard of useful science is its ability to make predictions and then to have those predictions be verified in experiment!). But it took a few years before the model made sense.
**Obviously, from a modern perspective, this was too optimistic (see the Uncertainty Principle), but this is a world before Quantum Mechanics, so I don't care.
***No room, or time, to explain why, except to note that if like charges repel and opposite charges attract, then why don't the electrons fall into the nucleus?