Going over Newtonian theory, energy and momentum are conservative, for one particle object, we have the below equation, in which, K represents kinetic energy, approximated as m c square plus the famous 1/2mv^2.
The gravitational force is similar to electric force
The U here denotes the concept of potential energy, then we deduce and understand the concept of potential, or more specific, electric potential. Think of it as the potential to have potential energy should you put a test charge in the system, or the gh, (constant g times height) is the potential to fall due to gravity should you put a test object to somewhere relative to the earth ground.
Getting into an area full of electric charges is analogous to be in a mountain of different level of height, stepping up or down, one needs to calculate to battle the environment.
Magnetism later on is discovered as well as its interaction to electricity. Starting the research form observing a point charge, Biot-Savart law is deduced:
Note the right hand rule is applied to judge the direction of the magnetic field B, flipping around, so can be used to judge the direction of current/charge velocity.
Manifested on math level:
As to the current in a wire when it’s connected,
There are two useful gadgets in circuit: capacitor and resistor.