There are lots of laws pertaining to electric bicycles. Some are precise while others aren't. Here's the definition of electric bicycle. Note that it could be argued that a regular bicycle with an added electric motor would not fall into this category, thus making it just a bicycle and having only to follow normal bicycle laws unless defined elsewhere.
CHAPTER 541. DEFINITIONS
(24) "Electric bicycle" means a bicycle that:
(A) is designed to be propelled by an electric motor, exclusively or in combination with the application of human power;
(B) cannot attain a speed of more than 20 miles per hour without the application of human power; and
(C) does not exceed a weight of 100 pounds.
Note Chapters 541 and 551.
http://www.statutes.legis.state.tx.us/