In Anthem, why does Equality 7-2521 perform experiments to discover electricity?
Equality is unusual in the collectivist society of the future; he asks questions and is unhappy with his assigned role. When he discovers the collapsed subway tunnel, Equality finds that he enjoys remaining isolated from others in society, and later he discovers the power of electricity by accident. Because of the stigma associated with acting outside one's assigned roles and boundaries, Equality must perform his experiments in secret.
No single one can possess greater wisdom than the many Scholars who are elected by all men for their wisdom. Yet we can. We do. We have fought against saying it, but now it is said. We do not care. We forget all men, all laws and all things save our metals and our wires. So much is still to be learned! So long a road lies before us, and what care we if we must travel it alone!
(Rand, Anthem, noblesoul.com)
Equality cannot share his work with others, or he will be punished or even killed. His experiments prove the uncharted power of electricity; Equality has discovered that although the society claims that there is nothing unknown in the world, this is a lie. By experimenting with electricity, Equality proves to himself that the leaders of society are lying to him and to everyone; their lies are vital for controlling society and keeping people from developing individualism. Equality shows himself that he is capable of great inventions alone, without help, and so continues his journey to individualism.