According to latest modern theory on subatomic particles, electrons and protons are further divided into quarks, having fractional charges.
My question is, why can't they exist independently? and why don't they show up Millikan's experiment?
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Hadrons are held together by the strong forces between quarks. These forces, in contrast to the electromagnetic ones which fall with distance as 1/r^2 (and thus allow us to detect free electrons, whose potential falls like 1/r), they behave like springs : potential proportionally to r , i.e. the larger the distance the force does not diminish , so as to allow freedom for quarks,( for energies within our every day laboratory experiments). Thus there can be no free quarks for a Millikan oil drop experiment.
At very high impact energies the potential is different and acts effectively also as 1/r , as the other forces, but the quarks may become free only to form a quark-gluon plasma. This is a hypothesis being tested currently in experiments at the LHC.