The reason that protons do not repel each other strongly enough to cause the nucleus to disintegrate is due to the strong nuclear force, also known as the strong interaction or strong nuclear interaction. The strong force is one of the fundamental forces of nature and is responsible for holding the atomic nucleus together.
The strong force is incredibly powerful at very short distances, acting between nucleons (protons and neutrons) within the nucleus. It overcomes the electromagnetic repulsion between protons, which arises from the positive charges carried by the protons. The electromagnetic force, which governs interactions between charged particles, is repulsive between protons due to their like charges.
However, the strong force has a much shorter range than the electromagnetic force. It becomes significant only within the range of about 1 to 3 femtometers (10^(-15) to 3 x 10^(-15) meters). Within this range, the strong force is attractive and can overcome the electromagnetic repulsion, allowing protons to be held together in the nucleus.
This does not violate Pauli's exclusion principle, which states that no two identical fermions (particles with half-integer spin, such as protons and neutrons) can occupy the same quantum state simultaneously. The exclusion principle applies to particles within the same energy level and orbital, ensuring that they have different quantum numbers. In the context of the nucleus, protons and neutrons occupy different energy levels and orbitals, so they can coexist in the nucleus without violating the exclusion principle.
It's important to note that the balance between the strong force and electromagnetic repulsion is delicate, especially in larger nuclei with more protons. As the number of protons increases, the electromagnetic repulsion becomes stronger, requiring additional neutrons to stabilize the nucleus. The interplay of these forces determines the stability and properties of different atomic nuclei.