The cost of charging your electric car at a public charge point depends on the charge point network and the location of the charge point. Many local authorities offer a pay per session approach to on street chargers. Occasionally they can be free to use if you have access to a network subscription.
Rapid charge points are typically found at motorway service stations and can also be free for certain drivers but are generally seen as one of the more expensive options. In essence, because they offer a faster charge (drivers to typically charge an electric car to 80% in 20-40 mins) and greater convenience, they tend to come at a premium.
However, if you are a Tesla owner, then the Tesla Supercharger Network has points across the UK which are often free to use. Newer Tesla owners, who bought their vehicle after January 1st 2017, may only receive a set number of free hours of charge. Please consult the manufacturer if you are in doubt.