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Charged Particles in Electric Fields

A charge q in a field E feels a force F = qE, so it has acceleration a = qE/m. Released from rest through a potential difference U it gains kinetic energy qU, giving v = √(2qU/m). Shot in across the field, it follows a parabola, like a thrown ball, with deflection y = qUL²/(2mdv₀²). A cathode-ray oscilloscope uses this to move an electron beam on a screen.

🎬 Step-by-step story

  1. Two flat plates: the top is + and the bottom is −. Between them the field E = U/d is the same everywhere and points from + to −. A charge placed here feels a force.
  2. Release a + charge near the top plate. The force along the field makes it speed up toward the − plate: a = qE/m. After crossing the voltage U its kinetic energy is qU.
  3. Make U four times bigger. The charge now arrives twice as fast. Since ½mv² = qU, speed grows as the square root of U.
  4. Now shoot the charge in sideways across the field. Sideways speed stays the same, but the force keeps pulling it down: a curved, parabola-shaped path. Afterwards it flies straight and lights the screen.
  5. An oscilloscope sends a steady beam of electrons between the plates. As the voltage swings up and down, the beam swings too, and the bright spot moves on the screen.
  6. Free play: change the sign of the charge, the voltage U and the entry speed. Try releasing it or shooting it across.

Tip: drag the 3D scene to turn it. Use two fingers to zoom.

🤔 Common doubts, cleared

Why does the charge speed up along the field but not stay at a steady speed?

The force qE acts all the time, so there is constant acceleration. In the 3D the particle moves farther every second as it falls.

Why does the speed depend on the voltage but not on the gap between the plates?

The work done is qU whatever the gap. A wider gap weakens the field but gives a longer path, and these cancel.

Why does the charge curve instead of going straight when shot sideways?

The sideways speed stays the same, but the field keeps adding downward speed. Together they make a curved path, the same as a thrown ball under gravity.

Why does the path become straight after the plates?

Outside the plates there is no field, so no force. The particle keeps its last velocity and goes in a straight line to the screen.

How does the oscilloscope show a signal as a picture?

The signal voltage moves the beam up and down. Because the spot's displacement is proportional to that voltage, the screen shows the signal like a graph.

What happens if the voltage is too large?

The deflection grows until the particle hits a plate. Try a high U and a low entry speed in the free play and watch the warning appear.

Force and acceleration in a uniform field

Between two parallel plates with potential difference U and gap d, the electric field is uniform: E = U/d, pointing from the + plate to the − plate. A particle with charge q feels the force F = qE. A + charge is pushed along the field; a − charge (such as an electron) is pushed against it.

By Newton's second law the acceleration is a = qE/m. It is constant, so the usual equations of motion work. An electron's mass is about 1800 times smaller than a proton's, so it accelerates about 1800 times more for the same field. In these problems gravity is tiny compared with the electric force and is ignored.

Acceleration along the field: speed gained

A charge q starts from rest and moves through a potential difference U. The work done by the field is qU, and it becomes kinetic energy:

½ m v² = q U, so v = √(2qU/m).

The result depends only on the voltage crossed, not on the gap d. Double U and the speed rises by √2; make U four times bigger and the speed doubles. The unit electron-volt (1 eV = 1.6 × 10⁻¹⁹ J) is the energy of an electron after crossing 1 V. A proton through 1000 V gains 1000 eV.

Deflection across the field

Now shoot the particle in with speed v₀ at right angles to the field, through plates of length L. This is exactly like horizontal projectile motion:

Along the plates: no force, so constant speed v₀, and the time inside is t = L/v₀.
Across the plates: acceleration a = qU/(md) from rest, so the sideways deflection is y = ½ a t² = qUL²/(2 m d v₀²), and the sideways speed on leaving is vy = a t.

The path inside the plates is a parabola. After leaving the plates there is no force, so the particle moves in a straight line at angle θ with tan θ = vy/v₀ = qUL/(m d v₀²). If y becomes bigger than half the gap, the particle hits a plate.

The cathode-ray oscilloscope

An oscilloscope tube has four parts. (1) The electron gun: a heated cathode gives off electrons, and an anode at high voltage speeds them into a narrow beam. (2) Vertical (Y) plates: the signal voltage is applied here and bends the beam up or down. (3) Horizontal (X) plates: a saw-tooth "time-base" voltage sweeps the beam steadily from left to right and snaps it back. (4) A fluorescent screen glows where the beam lands.

For a fixed accelerating voltage, the spot's displacement on the screen is proportional to the voltage on the plates (Y ∝ U). So the screen draws a graph of voltage against time. Electrons are used because they are light, so even small voltages bend them a lot, and they respond almost instantly to fast signals.

Try it at home

Roll a marble across a table while a friend tilts the table to one side. The marble keeps its forward speed but curves toward the low side, a parabola, just like a charge crossing a field. Then roll it straight down the tilt: it only speeds up. These are the two cases in the 3D.

Key formulas and definitions

Worked examples

1. A proton starts from rest and is accelerated through 1000 V. Find its speed. (e = 1.6 × 10⁻¹⁹ C, m = 1.67 × 10⁻²⁷ kg)

v = √(2qU/m) = √(2 × 1.6×10⁻¹⁹ × 1000 / 1.67×10⁻²⁷) = √(1.92×10¹¹) ≈ 4.4 × 10⁵ m/s.

2. An electron starts from rest and moves through 100 V. Find its speed. (m = 9.11 × 10⁻³¹ kg)

v = √(2 × 1.6×10⁻¹⁹ × 100 / 9.11×10⁻³¹) = √(3.51×10¹³) ≈ 5.9 × 10⁶ m/s.

3. Two plates 2 cm apart have 200 V across them. Find the field and the acceleration of an electron between them.

E = U/d = 200/0.02 = 10⁴ V/m. a = eE/m = 1.6×10⁻¹⁹ × 10⁴ / 9.11×10⁻³¹ ≈ 1.76 × 10¹⁵ m/s².

4. An alpha particle (charge 2e) goes through 500 V from rest. Find its kinetic energy in joules and in eV.

KE = qU = 2 × 1.6×10⁻¹⁹ × 500 = 1.6 × 10⁻¹⁶ J, which is 1000 eV.

5. An electron enters plates of length 5 cm with v₀ = 2 × 10⁷ m/s. The sideways acceleration is 3.5 × 10¹⁴ m/s². Find the time inside and the deflection.

t = L/v₀ = 0.05/(2×10⁷) = 2.5 × 10⁻⁹ s. y = ½ a t² = 0.5 × 3.5×10¹⁴ × (2.5×10⁻⁹)² ≈ 1.1 × 10⁻³ m = 1.1 mm.

6. A proton and an alpha particle are released from rest through the same voltage. Compare their speeds (alpha: charge 2e, mass about 4 times the proton).

v ∝ √(q/m). For the alpha, q/m is 2/4 = 1/2 of the proton's. So vα/vₚ = √(1/2) ≈ 0.71. The alpha is slower, but its kinetic energy (qU) is twice as big.

Common mistakes

Practice quiz

1. An electron is placed between a + plate (top) and a − plate (bottom). It moves:
2. A charge falls through voltage U from rest. Its kinetic energy is:
3. If U is made 4 times bigger, the final speed (from rest):
4. A charge enters a uniform field at right angles to it. Its path inside the field is a:
5. In an oscilloscope the spot on the screen moves up when:

Practice: answer these yourself

Type or choose your answer, then press Check. Use a hint if you are stuck; the full solution appears after you answer.

Frequently asked questions

How does a charged particle move in an electric field?

It feels the force F = qE. Released from rest or moving along the field, it speeds up in a straight line. Moving across the field, it follows a parabola.

How does an oscilloscope work in simple words?

A gun fires a beam of electrons. Plates bend the beam up and down with the signal and sweep it left to right with time. The beam lights a screen, so the spot draws the signal as a graph.

Why are electrons used in oscilloscopes?

They are very light, so even a small voltage bends them a lot, and they react almost instantly to fast signals.

Where this is taught

China高一Compulsory 3 Ch.10 Energy in electrostatic fields

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