A Charge in the Electric Field of a Capacitor: Why the Path Is a Parabola

Between the two plates of a capacitor there is a uniform electric field. It keeps pushing a charged particle to one side, while along the plates nothing slows it down — so the path is a parabola, like a thrown ball. Change the voltage, the speed and the particle itself and see whether it gets out, at what angle, or crashes into a plate. There are the “Oscilloscope” and “Droplet hovers” experiments, a quiz mode and a real Millikan experiment.

t where the particle is now (scrub)

→ v — velocity of the particle → F — force from the field → E — the field (from “+” to “−”) → mg — gravity plates: red is “+”, blue is “−” trail (blue → orange: slow → fast) ● marks at equal time intervals

The idea in 30 seconds

The field pushes the charge

Between the plates the field is uniform: the same at every point. A charge feels the force F = qE, so it gets the acceleration a = F/m. The sign of the charge picks the direction: “+” goes along the field, “−” against it; the mass decides how easy it is to budge the particle.

Two motions — independent

Along the plates nothing pushes: the particle flies at a constant speed vₓ. Across them a constant force acts: v_y grows like for a ball thrown horizontally off a table. Switch on the stroboscope: at equal time steps the dots move the same distance to the right, but up (or down) farther and farther (1 : 3 : 5 : 7…).

Parabola and energy

Adding the two motions gives a parabola. The field does work A = qΔU, and the kinetic energy grows by exactly that much. The deflection Δy = qUl²/(2mdv₀²) grows with U and l and falls with mass and speed — heavy and fast particles hardly notice the field.

Try it yourself

Press “Reverse polarity”: the force turns around and the trajectory mirrors itself.
Pick “Proton”. The same charge but 1836 times the mass — the parabola almost turns into a straight line.
Lower the entry speed v₀: the deflection grows like 1/v₀². Find the speed at which the particle crashes into a plate.
Switch on “Screen” and “Beam” — this is an oscilloscope: change U and the glowing spot runs across the screen.
The “Droplet hovers” experiment: include gravity and tune U so that qE = mg.

Where it is in real life

The oscilloscope and the picture tube of an old TV: two pairs of plates deflect an electron beam — that is how the image is drawn.
An inkjet printer: ink droplets are charged and deflected by a field so that they land at the right spot on the paper.
Electrostatic precipitators on factory chimneys and electrostatic painting: charged particles fly to a plate along the field.
Millikan’s experiment: from the balance of a droplet in the field the charge of the electron e = 1.6·10⁻¹⁹ C was measured — a droplet’s charge is always a multiple of e.

Graphs: the same in numbers

The curves are built from exact formulas. The bright part is what has already happened, the pale one is what will happen; the yellow dashed line is “now”, the white one is the moment of exit or impact.

Along the plates: equal distance in equal time. x(t) is a straight line

Across the plates: the distance per equal time grows. y(t) is a parabola

Speeds: vₓ stays the same, v_y grows along a straight line

Energy: kinetic grows at the expense of potential, the sum stays put

Mode

Quick experiments

Particle

q sign of the charge −

U voltage (how charged the plates are)

d gap between the plates

l plate length

v₀ entry speed

α entry angle 0°

y₀/d entry height (fraction of the gap) 0.50

Playback speed

Physics

Start from rest (v₀ = 0)

Include gravity

Show

stroboscope (equal time steps)

potential and field between the plates

screen (like in an oscilloscope)

beam (many particles in a row)

the hint “as if it came from the middle”

Any change of the parameters restarts the experiment. The motion is computed from exact formulas, so the numbers do not depend on the frame rate or on the “playback speed”. Edge effects at the ends of the plates are ignored.

KEY FORMULAS

Motion along and across the plates

$$x = v_{0x}\,t, \qquad y = v_{0y}\,t + \frac{a t^2}{2}, \qquad a = \frac{qE}{m} = \frac{qU}{m d}$$

No force along the plates — uniform motion; a constant acceleration from F = qE acts across them

▶ Detailed Explanation

The uniform field E = U/d acts perpendicular to the plates, so the acceleration a = qE/m = qU/(m·d) points across the plates and is constant in time. There is no force along the plates, so the horizontal component of the speed stays unchanged: x = v₀ₓ·t. Across the plates the particle accelerates uniformly, the coordinate grows parabolically y = v₀ᵧ·t + a·t²/2 and the speed vᵧ = v₀ᵧ + a·t changes linearly. This is why the y(t) graph is a parabola while the vᵧ(t) graph is a straight line of slope a.

Acceleration is inversely proportional to mass: in the same field an electron is kicked thousands of times harder than a proton, while a heavy ion or droplet is barely deflected.

Flight time and deflection at the exit

$$t = \frac{l}{v_{0x}}, \qquad \Delta y = \frac{q\,U\,l^{2}}{2\,m\,d\,v_{0}^{2}}$$

For entry along the plates (v₀ᵧ = 0): the particle covers the length l uniformly and is deflected by Δy

▶ Detailed Explanation

The time in the field is fixed by the horizontal component alone: t = l/v₀ₓ. In that time the vertical speed reaches vᵧ = a·t and the deflection is Δy = a·t²/2 = q·U·l²/(2·m·d·v₀²). This gives the key practical rule: the deflection grows with the square of the plate length and with the voltage, and shrinks with a larger gap d and with the square of the entry speed.

If the computed Δy exceeds the distance to the nearest plate (d/2 for entry in the middle), the particle never reaches the exit — it strikes a plate. The simulator compares both times, the flight time and the impact time, and treats the earlier one as the event.

Exit angle from the capacitor

$$\tan\varphi = \frac{v_{y}}{v_{x}} = \frac{q\,U\,l}{m\,d\,v_{0}^{2}}$$

φ is the angle between the exit velocity and the direction of the plates

▶ Detailed Explanation

At the exit the velocity vector is the sum of the unchanged vₓ = v₀ and the gained vᵧ = a·t, so tg φ = vᵧ/vₓ = q·U·l/(m·d·v₀²). The angle grows with the voltage and the plate length and falls with the square of the speed — a very fast particle follows an almost straight line, while a slow one can turn through nearly 90°.

The exit point is found the same way: y = y₀ + Δy. If it falls outside the gap, the particle hits a plate, and then no exit angle exists — the motion stops at the point of impact.

Spot shift on an oscilloscope screen

$$Y = \left(L + \frac{l}{2}\right)\tan\varphi = \left(L + \frac{l}{2}\right)\frac{q\,U\,l}{m\,d\,v_{0}^{2}}$$

The spot shift is proportional to the voltage U: the larger U, the farther the spot from the centre of the screen

▶ Detailed Explanation

After leaving the capacitor the particle flies along a straight line at the angle φ. Extended backwards, this line passes through the middle of the plates (x = l/2). So on a screen at a distance L from the end of the plates the shift is Y = (L + l/2)·tan φ. Substituting tan φ = qUl/(mdv₀²) shows that Y is proportional to U — this is the principle of the oscilloscope: the voltage is turned into the shift of a glowing spot.

In the simulator press “Oscilloscope”: the screen and the beam switch on. Change U and watch the spot run across the screen.

Work of the field and kinetic energy

$$A = qE\,\Delta y = q\,U_y = \Delta E_k, \qquad v = \sqrt{\frac{2qU}{m}}$$

The work of a uniform field equals the change in kinetic energy; the second formula is the acceleration from rest across the full voltage

▶ Detailed Explanation

The force F = qE is constant, so the work A = F·Δy = q·E·Δy depends only on the displacement along the field and equals q·U_y, where U_y is the potential drop between the entry and exit points. By the work–energy theorem this work equals the gain in kinetic energy: ΔEк = A = q·E·Δy. On the Eк(t) graph the curve grows as the square of the time in the field — the same parabolic law as in y(t).

Acceleration from rest is the special case: a particle starting at the positive plate sweeps the full potential drop U and gains ½·m·v² = q·U, hence v = √(2qU/m). This is how electrons are accelerated in cathode ray tubes and electron guns. With gravity included, the polarity can be chosen so that qE = mg: the resultant force vanishes and the particle travels in a straight line — this is the basis of Millikan’s experiment on measuring the elementary charge.

Droplet equilibrium: the Millikan experiment

$$q\,E = m g \;\Rightarrow\; q = \frac{m\,g\,d}{U}, \qquad q = n\,e$$

When the field force balances gravity the droplet hovers; the hovering voltage gives its charge, and it is always a multiple of e

▶ Detailed Explanation

If the field force points up and equals the weight, the net force is zero and a droplet released from rest stays where it is. From qE = mg with E = U/d we get q = mgd/U. Millikan measured the masses of droplets and the hovering voltages and found that the charges always come out as an integer multiple n·e, where e = 1.6·10⁻¹⁹ C. This proved that electric charge is discrete.

In the “Millikan lab” mode the droplet hides the number n: tune U until it hovers, press “Find the charge” — and the simulator computes q and n from your U.

A parallel-plate capacitor is two parallel metal plates. Connect a voltage U to them and one plate becomes “+”, the other “−”, and between them a uniform electric field E = U/d appears, where d is the gap between the plates. “Uniform” means that at every point between the plates the field is the same — the same size and the same direction, from “+” to “−”. In the simulator it is drawn as violet arrows, and the scale on the left shows the potential φ: it changes smoothly from U at the “+” plate to 0 at the “−” plate.

A charged particle in this field feels a constant force F = qE. A positive charge is pushed along the field, a negative one against it. By Newton’s second law the acceleration a = F/m = qU/(m·d) is the same at every point between the plates. That is why a light electron accelerates 1836 times more than a proton with the same charge, while a heavy droplet hardly feels the field.

Split the motion into two independent ones. Along the plates there is no force, so the speed vₓ does not change and x = x₀ + v₀ₓ·t is uniform motion. Across the plates a constant force acts, so y = y₀ + v₀ᵧ·t + a·t²/2 and vᵧ = v₀ᵧ + a·t grows along a straight line. This is exactly like a ball thrown horizontally off a table, only the force of the field takes the place of gravity. Add the two motions and you get a parabola. Switch on the “stroboscope”: in equal time steps the dots move the same distance to the right, but up (or down) by more and more, like 1 : 3 : 5 : 7.

The flight time through the capacitor is t = l/vₓ. For entry along the plates the deflection at the exit is Δy = qUl²/(2mdv₀²), and the exit angle is tan φ = vᵧ/vₓ = qUl/(mdv₀²). The faster the particle, the less it can deflect; the heavier, the same. If the computed Δy is bigger than the distance to the nearest plate (d/2 for entry along the middle), the particle cannot get out and hits a plate. The simulator finds the moment of exit or impact by solving an equation, not by “the first frame that crosses the line”, so the numbers are exact.

Here is a neat fact: the exit velocity line, extended backwards, passes through the middle of the plates (x = l/2). Switch on the hint “as if it came from the middle” and check. That is why on a screen at a distance L from the end of the plates the spot shifts by Y = (L + l/2)·tan φ — this is how the deflection system of an oscilloscope or an old picture tube works: change the voltage U and the spot runs across the screen.

Energy. The field does work A = qEΔy = qΔU, and the kinetic energy grows by exactly that much: ΔEₖ = A. For particles a handy unit is the electronvolt: 1 eV = 1.6·10⁻¹⁹ J, the energy an electron gains after crossing a potential difference of 1 V. From rest across the whole voltage U a particle reaches v = √(2|q|U/m) — this is how an electron gun works. On the energy graph the kinetic energy grows, the potential energy falls, and their sum is a horizontal line: energy is conserved.

Gravity. For an electron it is 10¹⁴ times weaker than the field force — it is simply dropped. But for a microscopic oil droplet the forces are comparable, and the voltage can be tuned so that qE = mg: the droplet hovers, with q = mgd/U. In 1909 Millikan measured droplet charges exactly like this and saw that they are always multiples of one number — the electron charge e = 1.6·10⁻¹⁹ C. Switch on the “Millikan lab” mode: the droplet hides an unknown number of extra electrons — tune U until it hovers and find that number.

How to use it: press Start and watch the scene — the readout in the top right shows real time and speeds (the flight lasts nanoseconds, on screen it is stretched). The “Quick experiments” buttons set up interesting parameters for you. In the “Quiz” mode the path and the numbers are hidden: first predict which way the particle bends and whether it gets out, then watch the answer.