# 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? Feb 13, 2017 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. Feb 13, 2017 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 Aug 23, 2019 at 10:45
• That is zero if v=6 is constant with magnitude of test charge Q Aug 23, 2019 at 10:47

What you want is

$$\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}$$

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}

Potential $$(V)$$ is just potential energy pet unit charge

$$V=\frac{E_p}{q}$$

So yes, they only differ in a constant. If $$V$$ is the same, then the energy of the charge is also the same.

Take into account that $$V$$ exists anyways, while potential energy depends on the charge. Potential is something there's in the space, on all points, but potential energy is something that the system has.

If the system has total charque $$Q$$ and it is entirely contained in a constant potential region, then

$$E_p=Q\cdot V$$

In general, for a system of $$N$$ charges, the potential energy is

$$E_p=\sum_{i=1}^N q_i \cdot V_i(at\ the\ point\ of\ q_i)$$

Since potential is just work done or energy per unit charge, potential energy will also be same everywhere for a charge Q.

• so the electric potential energy will be the same everywhere right but would it be zero? or not ?
– Dee
Feb 13, 2017 at 14:10
• @Dee It can be zero, when charge is zero ie, field becomes zero. But constant potential energy means the change in potential energy will be zero, since its constant everywhere. Feb 14, 2017 at 16:08

The difference between electric potential and electric potential energy is that the potential refers to a region, but potential energy must be referring to some object. We can't really talk about the potential energy throughout the system, unless you're actually placing a charge at each point within the system. In that case, yes, the potential energy of a charge placed within an electric potential is given by

PE = qφ

where φ is the electric potential, so no change in potential means the charge would have the same energy no matter where you place it. This is assuming that placing the charge at that point doesn't have any significant effect on whatever is creating the electric potential.