Potential Energy of a point charge in uniform electric field

Question

I have to determine the potential electric energy from a point charge $q$ who is in an position $r$ inside an uniform electric field $\vec{E}$. I tried to do it by using work definite integral and then use $ V = \frac{U}{q} $, but i'm stuck because one of the integration limits is $\infty$, and the definite integral would look like: $ qrE - qE\infty $. The problem is that i don't know if this electric field goes to infinity (honestly, i think it does not) but i can't see other way of figuring this out. The correct answer (according to the book) is $qr · E$. Thanks.

Answer 1

The issue is that you may not take $\infty$ as a reference point since $\vec E$ is defined everywhere, say in the $\hat i$ direction to be $\vec E(x, y, z) = E_0\hat i$. Thus, when defining a potential $V(x, y, z)$ for this field, you'll have to choose a reference point, say $\vec r_0 = (x_0, y_0, z_0)$ and determine the potential at an arbitrary point $\vec r = (x, y, z)$ as $$ V(\vec r) = -\int_{\vec r_0}^{\vec r}d\vec s\cdot\vec E = -\int_{x_0}^xdx\,E_0 = E_0(x_0 - x). $$ In this case, the only meaningful quantities we can talk about are potential differences $\Delta V$ or potential with respect to a fixed point $\vec r_0$.

Answer 2

An infinite uniform electric field would be able to release an infinite amount of energy to a charge moving through it.

To get a finite result you'd either need to allow it to move only a finite distance through the field, or the field would have to be non-uniform.