Final answer:
Sir Humphry Davy was the first to demonstrate electrolysis of molten magnesium chloride in Downs' cell, and his assistant Michael Faraday furthered his work in electrochemistry.
Step-by-step explanation:
The first person in the laboratory to show the electrolysis of molten magnesium chloride in Downs' cell was Humphry Davy. Sir Humphry Davy was a pioneer in the field of electrolysis, where he used the voltaic pile to split common compounds. Through electrolysis of molten salts, he discovered several new metals and invented the field of electrochemistry. His laboratory assistant, Michael Faraday, enhanced Davy's work on electrolysis, becoming a fundamental contributor to the study of electromagnetism and electrochemistry. Faraday's advancements paved the way for technologies such as the electric motor and generator, as well as experiments using cathode ray tubes, which are crucial for the understanding of electricity today.