I'm having a bit of trouble understanding mass defect, in the context of nuclear physics. The argument from my class is:
The nuclear force is attractive, and so does work on the particles as they are brought from being far apart to being close together. I am happy with this
Since the force does work on the particles, the change in potential energy is negative. I wasn't happy with this, but I came across this which helped.
$E$ in the relationship $E = mc^2$ includes potential energy. I am happy with this.
The change in energy is therefore negative, so the total mass of the nucleons decreases after they are brought together. In other words, the mass of the nucleus is less than the mass of its constituent nucleons, when they are far apart.
It is point 4 I really don't understand. I don't see how it follows from points 2 and 3 at all. In thinking about my problem with this, I was constructing the following argument.
Consider two point particles, one of mass m and charge q, the other of mass m and charge -q. The masses here are when the particles are far separated.
In bringing the particles together, the change in their potential energy is negative. To me, this seems exactly like point 2 of the argument above.
The particles can get arbitrarily close together. If we are modelling the particles as points, then this is true.
There is a distance $d$ such that the difference between the initial potential energy of the particles and the potential energy of the particles when separated by distance $d$, $\Delta U$, has $\Delta U \leq -m c^2$.
Therefore, the particles end up with zero or negative mass, just by being allowed to attract each other.
Obviously, this is argument is not sound, but I can't see how it differs from the argument about the nucleus mass.
My initial thought in constructing this argument was that the particles gain kinetic energy as they attract each other. In particular, the kinetic energy the particles gain cancels out the potential energy the particles lose, and so there is zero change in energy. This doesn't really help me understand mass defect though, since surely the same could be said of the nucleons as they attract each other?
I think we would have to do some work on the nucleons to stop them from moving. In other words, there is no way to go from the state where the nucleons are at rest and far apart, to the state where the nucleons are at rest and bound together, without doing work on the nucleons.
Can anyone help me make sense of this?