A charge crossing a uniform field moves exactly like a ball thrown sideways under gravity: steady along the plates, accelerated across them. The same splitting as Horizontal Launches works. Each problem here is one short scene and a chain of questions. Answer them in order: each later question uses what you found before, so a slip early on carries through. Keep full precision between steps and round only at the end.
Key idea
$$E = \frac{V}{d}, \qquad a = \frac{qE}{m}, \qquad t = \frac{\ell}{v_0}, \qquad y = \tfrac{1}{2}at^2.$$ An electron is pulled toward the positive plate; gravity is far too weak to matter.
Worked example: An electron beam
An electron at $2.0 \times 10^7\ \text{m/s}$ enters plates $4.0\ \text{cm}$ long, $2.0\ \text{cm}$ apart, with $100\ \text{V}$ across them. Find (1) the field, (2) the time inside and (3) the deflection.
- (1) $E = 100/0.020 = 5000\ \text{V/m}$.
- (2) $t = 0.040/(2.0 \times 10^7) = 2.0 \times 10^{-9}\ \text{s}$.
- (3) $a = (1.6 \times 10^{-19})(5000)/(9.11 \times 10^{-31}) = 8.78 \times 10^{14}\ \text{m/s}^2$; $y = \tfrac{1}{2}a t^2 = 1.76\ \text{mm}$.
Answer: $5000\ \text{V/m}$, $2.0\ \text{ns}$, $1.76\ \text{mm}$
Common mistake
Using $V$ where $E$ belongs. The field is volts per meter: divide by the gap in meters. Check units: $qE$ must come out in newtons.