A charged particle in a uniform electric field undergoes uniformly accelerated motion. There are many applications, including Millikan’s experiment and the principle of the cathode-ray tube.
Basic Knowledge
Force acting on a charge q of mass m in a uniform electric field E:F = qE
Acceleration: a = qE/m
When entering perpendicular to the electric field, the particle undergoes projectile motion (initial velocity v₀ is horizontal, and the electric field produces vertical acceleration).x = v₀t (uniform motion), y = ½at² (uniformly accelerated motion)
Millikan’s oil-drop experiment:
When gravity and Coulomb force balance in electric field E:qE = mg → q = mg/E
This measures the elementary charge e (confirming that charge is an integral multiple of e).
Energy of a charged particle accelerated through voltage V:qV = ½mv² → v = √(2qV/m)
Projectile motion (The trajectory of a charged particle entering perpendicular to an electric field. It has the same form as projectile motion in mechanics)
Millikan’s experiment (Applying an electric field to an oil drop, balancing it against gravity, and measuring the elementary charge)
Accelerating voltage (Voltage used to accelerate charged particles. The speed is determined by qV = ½mv²)
Electron gun (A device that emits electrons from a cathode and accelerates them with accelerating voltage)
Cathode-ray tube (A display device that deflects an electron beam with an electric field)
Further Study (Background and Significance)
In projectile motion, motion in the direction of the electric field and motion in the perpendicular direction are independent. In the x direction: uniform rectilinear motion; in the y direction: uniformly accelerated motion. The trajectory is y ∝ x² (a parabola).
Millikan’s experiment (1909) not only precisely measured the elementary charge but also demonstrated the revolutionary fact that “charge is quantized.” The modern measured value is e = 1.602 × 10⁻¹⁹ C.
A mass spectrometer also uses the same principle: particles are accelerated by an accelerating voltage and bent by a magnetic field to measure the mass-to-charge ratio (m/q).
- F = qE → a = qE/m
- Perpendicular entry into an electric field → projectile motion
- Accelerating voltage V: qV = ½mv²
- Millikan: measure q using qE = mg
- Motion in and perpendicular to the electric-field direction is independent
- x: uniform motion, y: uniformly accelerated motion (parabola)
- A mass spectrometer uses the same principle
Points to Note (Easily Confused)
① The direction of acceleration in an electric field is reversed depending on whether the charge is positive or negative. ② For accelerating-voltage calculations, use qV = ½mv² (conservation of energy). ③ In Millikan’s experiment, you may also be asked to reverse the direction of the electric field to make the oil drop stationary. ④ Do not confuse this with the motion of charged particles in a magnetic field (circular motion).
Practice
- Find the acceleration of an electron in an electric field E = 1.0 × 10³ V/m, given m = 9.1 × 10⁻³¹ kg and charge e = 1.6 × 10⁻¹⁹ C.
- Find the speed of an electron accelerated through an accelerating voltage of 1000 V.
- In Millikan’s experiment, express the electric field E that balances an oil drop of mass m and charge q in terms of m, g, and q.