What is the significance of electric potential energy in electrostatics?

[kripkrip420]

Homework Statement

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Hi there! We are currently studying forces and fields in physics and we are in the electrostatics chapter. I am having a bit of trouble fully understanding electric potential energy. The book states that electric potential energy is defined as the work required to move a charge in the opposite direction of the electrostatic force. As I recall from previous chapters, work is defined as a CONSTANT force over a displacement. The problem with electric potential energy is that electric fields do not provide a constant force since they become weaker as a charge moves farther from a source charge. There are only two scenarios that I can think of where an electric field can have uniform magnitude and direction. If the source charge has infinite charge or if the test charge is placed between oppositely charged plates. The reference point also confuses me. If the reference point is at infinity, how is it possible to have this exist between two plates? There is no location between two plates where a charge can exist an infinite distance away(in my mind). The infinite charge scenario makes more sense but still, if there is infinite charge, then that must mean that the electric field generated must have infinite strength at ANY point or distance from the source. So how can an infinite distance possible result in a zero electric potential when an infinite charge results in an infinite field? All in all, electric potential energy ( at least the work definition) works only in fields that have constant magnitude and direction (uniform) correct?

Thank you for your help!

Homework Equations

The Attempt at a Solution

 

[jhae2.718]

Are you doing calculus or algebra-based physics?

If you are doing calculus based physics, work is defined as $$W = \int_{\mathbf{r_1}}^{\mathbf{r_2}} \mathbf{F} \cdot d\mathbf{r}$$. So, work doesn't require a constant force; only the expression W = |F||D|cos(θ) requires a constant force as it is the solution to the integral when F is constant.

Then, if you know the electric field, you can find the force exerted on a charge by the E field, and then use U = -W to find the electric potential energy.

For the case when the test charge is at infinity: the electric potential energy for a field given by Coulomb's Law is $$U(r) = \frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r}\hat{\mathbf{r}}$$. What does U(r) tend to when r is made very large?