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I believe that like charges (electric) do not attract each other.  This is according to the Coulomb's law of attraction or repulsion between electric charges.

Let there be two like charges on two parallel plates kept at a distance, then we  introduce some dielectric medium/slab like a glass which lets static electricity on it.  Then on the two surfaces of the slab static charges are accumulated.  They will be opposite in sign to the charges on parallel plates.  This is due to electro static induction.
     Then in that case,  the two parallel plates move towards each other. But it is not due to attraction between the plates, it is due to attraction between the charges and the induced charges.

In magnetism too, like poles repel each other.

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In most of the situations, like charges repel while unlike charges attract. This is a perfect rule. But, when the charges are a lot microscopic and distance between them is around an order of 0.6 Fermi, they attract each other depending upon their spin orientation. This is what that happens between protons in the nucleus of atoms. The force that governs their attraction or repulsion is termed as "Strong Nuclear Force". In terms of magnitude it is stronger than electromagnetic force by a 100 times.
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