Charges Moving in Electric Fields

A uniform field is simply one that has the same strength and direction everywhere in a region, like the field between two big charged plates; "uniform" just means it doesn't change from place to place. It's what lets oscilloscopes, inkjet coders and particle accelerators steer charges predictably. All we're really saying here is that a charge in such a field moves just like a ball thrown through the air, with the field doing the job gravity usually does. The formula $\mathbf{a} = q\mathbf{E}/m$ is only three pieces: $q$ (how much charge, which sets how hard it's pushed), $\mathbf{E}$ (the field, which sets the direction) and $m$ (the mass, since heavier things are harder to speed up). Push divided by mass gives acceleration, which is Newton's second law and nothing new. If you've done projectile problems, you already know most of this lesson.

A uniform field. Two large plates carry charge densities $+\sigma$ and $-\sigma$. As you found in Fields of Continuous Charge, each makes a field of $\sigma/2\varepsilon_0$, away from the positive plate and toward the negative one. Between the plates both point the same way and add to $\sigma/\varepsilon_0$; outside they cancel. So in the gap the field is the same everywhere. With a voltage $V$ across a gap $d$, it's $E = V/d$ (the volt arrives in Electric Potential).

Constant force, constant acceleration. The force $q\mathbf{E}$ is the same at every point, so $\mathbf{a} = q\mathbf{E}/m$ is constant. And it can be huge. For an electron $e/m_e = 1.76\times10^{11}$ C/kg, so a modest 1 kN/C field gives $1.76\times10^{14}$ m/s², more than $10^{13}$ times $g$. So you can ignore gravity for electrons and ions. For charged dust and ink drops, which are far heavier, compare $qE$ with $mg$ first.

The projectile analogy. You've seen constant acceleration before: it's what gravity does near the ground. A ball thrown sideways falls in a parabola, because gravity accelerates it steadily downward while its sideways velocity stays put. An electron entering parallel plates at speed $v_0$ does exactly the same thing, with "down" meaning the direction of $q\mathbf{E}$. Every projectile formula you know carries over:

After $t = L/v_0$ it leaves the plates with sideways velocity $v_y = aL/v_0$, at an angle given by $\tan\theta = aL/v_0^2$. Past the plates there's no field, so it flies in a straight line.

Where it lands. Its height at the exit is $y_1 = \frac{aL^2}{2v_0^2} = \frac L2\tan\theta$. So the straight exit path, extended backward, crosses the axis exactly at the plates' midpoint, and the spot on a screen a distance $D$ past the plates sits at

So you can pretend the electron gets a sudden kick at the middle of the plates and flies straight from there. On a typical tube, most of the deflection actually happens during that field-free drift: the plates set the angle, and the long trip to the screen turns it into distance.

Energy. The field does work $W = q\mathbf{E}\cdot\Delta\mathbf{r}$ (a dot product, the constant-force case of work): only motion along the field counts. Released from rest and moving a distance $d$ along the force, a charge gains $\frac12mv^2 = |q|Ed$, so $v = \sqrt{2|q|Ed/m}$. Crossing deflection plates is different. The electron moves along the field only by its small sideways deflection, so the plates steer it while barely changing its speed.

A common mistake. It's tempting to use $y = \frac12at^2$ for the whole trip to the screen, with $t$ the whole flight time, since that's how a thrown ball works. It isn't the same here: the field only exists between the plates. The acceleration lasts just $L/v_0$, and after that the path is a straight line. Two more things to watch. Electrons accelerate opposite to $\mathbf{E}$, so check your signs. And these formulas are only good below about a tenth of the speed of light, $3\times10^7$ m/s; past that you need relativity.

With a magnetic field instead, the force stays perpendicular to the velocity and the parabolas become circles (Magnetic Force on Moving Charges); crossing electric and magnetic fields builds a velocity selector.

Continuous inkjet coders print dates and batch numbers on bottles, cans and eggs by charging each drop and steering it between deflection plates. Electron-beam lithography writes chip patterns with deflected electron beams, and ion implanters steer charged atoms into silicon the same way. Linear accelerators, from hospital radiotherapy machines to particle colliders, push electrons and protons to high energy with strong fields along their path. Recycling plants sort shredded plastics with electrostatic separators, which deflect charged fragments as they fall between high-voltage plates.