3. (a) That's incorrect. Because battery is based on the notion of trespass to the person it is not necessary for a plaintiff to prove that they have suffered physical harm as a result of the unlawful contact. It is the act of interference with another's person that is actionable.

Therefore A could sue B for battery for the bump in the passage, even if the contact caused no injury. A would perhaps ask for an injunction to prevent similar conduct in the future.