Objectives

• To understand what the strong force is and what it applies to
• To appreciate that the Strong force is a short-range nature force
• To know that the Strong force is attractive to ranges of approximately $3 fm$ and repulsive at distance below $0.5 fm$.

Having knowledge on the general structure of the atom should now pose questions, especially if you have already studied the electric fields and electromagnetism. If protons and neutrons exist in the nucleus, the electrostatic force between protons should repel them from one another forcing them to separate and for the atoms to rip apart. This clearly does not happen as the world around us is made up of atoms which are not torn apart! So what is holding the protons (and neutrons, together?

The Stong Force

Evidence for the required existence of the strong force can be seen using Coulombs force equation;

$F = \frac{Q_{1}Q_{2}}{4\pi\epsilon_{0}r^{2}}$

Using this equation and the relative sizes of the atom, stated on the previous page the force each proton exerts on each other can be deduced;
$Q_{1} = 1.6 \times 10^{-19} C$
$Q_{2} = 1.6 \times 10^{-19} C$
$r = 1 fm = 1 \times 10^{-15} m$

Subbing these into Coulomb’s equation gives $F_{electrostatic} = 230 N$ (2 s.f.)
This is absolutely huge for two tiny particles that are not visible to the human eye!
Let’s say that they are gravitationally attracted to one another – perhaps this force can counteract this massive electrostatic force?

Using the Gravitation force equation;

$F = \frac{Gm_{1}m_{2}}{r^{2}}$

Using this equation and the relative masses of the atom;
$m_{1} = 1.275 \times 10^{-27} kg$
$m_{1} = 1.275 \times 10^{-27} kg$
$r = 1 fm = 1 \times 10^{-15} m$

Subbing these into the gravitational force equation gives $F_{gravitational} = 1.4 \times 10^{-34} N$ (2 s.f.)

So according to these results, the gravitational force is not enough to force the protons together.
Perhaps, let’s just for a second propose that maybe the electrons orbiting the nucleus are influencing the protons so much that they stay together?
Using the electrostatic force again, and the relative sizes of an atom as stated on the previous page;

$F = \frac{Q_{1}Q_{2}}{4\pi\epsilon_{0}r^{2}}$

$Q_{1} = 1.6 \times 10^{-19} C$
$Q_{2} = 1.6 \times 10^{-19} C$
$r = 1 fm = 0.5 \times 10^{-10} m$

Subbing these into the Coulomb’s equation gives $F_{electrostatic} = 9.2 \times 10^{-8} N$ (2 s.f.)

This is still no where near enough to keep the protons together, plus the electrons existing in a ‘cloud’ would likely results in this force cancelling out from when the electrons are on one side of the nucleus to the other – in other words, we should ignore this altogether!

The net force between two protons should be zero if it isn’t being forced apart;

$F_{net} = 0 \ N$
And we know;
$F_{electrostatic} = 230 \ N$
\$latex F_{gravitational} = 1.4 \times 10 ^{-34} N

So; $F_{net} \neq F_{electrostatic} + F_{gravitational}$

There must be some other force, which must be incredibly strong at short distances overcoming this difference in forces, the strong force, $F_{strong}$ we can therefore propose;

$F_{net} = F_{electrostatic} + F_{gravitational} + F_{strong}$

The strong force must only work at short distances because we only have elements up to a certain size in the periodic table, also, matter all around us is not all fused together!

The strong force also must vary with distance. This is evident because protons may not be as close to one another as shown in the example above, of two protons – this could be an example of Deterium ($^{2}_{2}H$, 2 protons but no neutrons). If we take an isotope of helium, $^{3}_{1}He$, the protons could be separated by a neutron, therefore (in the extreme case) the two protons could be separated by up to $2 fm$, this is double the distance apart from the previous example. This would mean that the electrostatic force would reduce to $58 N$ (which is still incredibly high). The gravitational force of anything would increase because there are now three nucleons involved instead of two a but the magnitude of this would still be negligible! The strong force cannot therefore be inverse square law relationship with distance.

What if the particles were forced together so they were the strong force becomes bigger than the electrostatic force?

If the protons got so close, once could think that the strong force may become so strong that it permanently attracts the two protons together into a singularity. This fortunately does not happen, as is evident by the fact the matter does not collapse in on itself when compressed!
The strong force, at even closer distances than $1 fm$ , becomes a repulsive force.

What does the strong force act upon?
The strong force acts between two particles that have a mass and are positioned typically $< 3 fm$ apart. The particles do not need to be charged, therefore the strong force can act between two neutrons or between a proton and a neutron.