# Why is the energy needed to move a particle between any two points independent of the path it takes

Say I had a charge +Q and I moved closer to a charge +100Q why is the energy needed independent (if I went some strange shape or a straight line) of the path +Q takes as long as it starts and ends at the same place. Many thanks!

• This is true if and only if the forces acting on the particles are conservative forces. Can you be a bit more specific about what you want to know about that? Dec 28, 2016 at 13:41

This is stated in the Maxwell Equation (in electrostatic) about the curl of the electric field: $$\nabla \times E = 0$$ with $E$ electrostatic field and $\nabla$ vector differential operator. This equivalent to say that the electrostatic field can admit a potential $\phi$ and there is the following relation: $$E = - \nabla \phi.$$ Stated this, you can prove that every path integral of a conservative field depends only on the initial and final position of the path, as stated in the Gradient theorem.