A charge of #2 C# is at the origin. How much energy would be applied to or released from a # -1 C# charge if it is moved from # (7 , -4 ) # to #(-3 , 5 ) #?
There is a very similar question here
This is spherically symmetrical, and:
#hat r#is the unit radial vector #(vec r )/(abs ( vec r))#.
#r = abs ( vec r)#, ie the distance of the point from the Origin
The change in potential [ie electrostatic potential energy per unit charge] in moving a unit positive test charge via any path from
The negative sign reflects that fact that the field is not doing the work, rather the work is being done against the field....
But also true is
So we can write
The change in energy of a charge
Reality Check : Because the negatively charged particle has moved closer to the positively charged Origin, the particle has lost potential energy. Tick.