Which sentence is correcty puctuated with a semicolon?

Like commas, semicolons indicate an audible pause — slightly longer than a comma's, but short of a period's full stop.

Rules for Semicolon ::

1. A semicolon can replace a period if the writer wishes to narrow the gap between two closely linked sentences.

2. Use a semicolon before such words and terms as namely, however, therefore, that is, i.e., for example, e.g., for instance, etc., when they introduce a complete sentence. It is also preferable to use a comma after these words and terms.

3. Avoid a semicolon when a dependent clause comes before an independent clause.

4. Use a semicolon to separate units of a series when one or more of the units contain commas.

Hence according to rule no. 2) option (C) He ate; however, he didn't sleep is correctly puctuated with a semicolon.