# Electric potential and electric potential energy relationship

I have a problem in where the electric potential has a constant value of six volts everywhere in a 3D region, the points are spread out. My question is would the electric potential energy of the system be the same all throughout no matter the distance since the electric potential is the same at all points?

• What system as there can be no charges in a region where the potential is constant? – Farcher Feb 13 '17 at 8:06
• $$\mathbf{E}=-\nabla \phi$$ As long as the test charge is within the region, the charge experiences no electrical force and hence no transfer of electrical energy takes place. – Ng Chung Tak Feb 13 '17 at 12:47
• There is a thing which you need to take an account ,what you refer as a reference point ,for calculation potential energy of V=6 change in potential energy is same – yuvraj singh Aug 23 at 10:45
• That is zero if v=6 is constant with magnitude of test charge Q – yuvraj singh Aug 23 at 10:47

$$\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}$$
The energy you might be looking for is: \begin{align} E &= \epsilon_0 \int\ \mathbf{E} \cdot \mathbf{E}\ \mathrm dV \\ &= \epsilon_0 \int\ \left(\nabla \phi + \frac{\partial \mathbf{A}}{\partial t}\right)\left(\nabla \phi + \frac{\partial \mathbf{A}}{\partial t} \right)\ \mathrm dV \end{align}