# Conservation of energy in a uniform electric field

Say you place a positively charged particle between two plates with an uniform electric field between them. The particle will accelerate towards the negatively charged plate with a constant acceleration, and the work done on the particle by the field is defined by:

$$\Delta W = qE \Delta x$$

Where $\Delta W$ is the work done, $q$ is the particle's charge, $E$ is the uniform field strength and $\Delta x$ is the distance the particle has moved.

From my understanding, $\Delta W$ should be equivalent to the particle's loss of absolute electrical potential energy and it's gain in kinetic energy. However, as an answer to a textbook question with this setup, $\Delta W$ was defined as the gain in EPE and also the gain in KE.

However, isn't that impossible, because if the particle gains $\Delta W$ in KE and in EPE then it will gain $2\Delta W$ in overall mechanical energy, which is impossible because only $\Delta W$ work has been done?

Is there an error in my textbook, or am I making a conceptual error somewhere?

• The particle loses EPE and gains KE. It does not gain both. – sammy gerbil Jul 26 '17 at 12:23
• That's what I thought. So then the textbook has an error. – Pancake_Senpai Jul 26 '17 at 13:02
• If the textbook says that both EPE and KE increase in this situation then Yes the textbook has an error. – sammy gerbil Jul 26 '17 at 13:06

In electrostatics, the work done on the charge in moving it from position $A$ to position $B$ is
Here, we choose the mechanical force $Fm$ such that it balances the electric force $qE$ at each point along the path. What does this say about the kinetic energy?