PRINCIPLE OF MILLIKAN’S DETERMINATION OF THE ELECTRONIC CHARGE

1. Millikan’s oil drop experiment:

  • Millikan’s oil drop experiment was a groundbreaking scientific study performed by Robert Millikan in 1909. The experiment aimed to measure the charge of a single electron, which was a fundamental constant in physics.
  • Figure 1 Oil drop experiment
  • Apparatus:
  • – A vacuum chamber
  • – A series of parallel metal plates (the “condenser”)
  • – A fine mist of oil droplets
  • – An electric field generator
  • – A microscope
  • Procedure:
  • – Oil droplets were introduced into the vacuum chamber.
  • – The electric field generator created an electric field between the metal plates.
  • – The oil droplets became electrically charged due to friction with the air.
  • – The microscope was used to observe individual oil droplets.
  • – By adjusting the electric field strength, Millikan was able to suspend individual oil droplets in mid-air.
  • – The suspended droplets were then observed, and their movements were tracked.
  • – The charge on each droplet was calculated based on its movement and the known electric field strength.
  • Millikan measured the charge on individual oil droplets and found that they were all integer multiples of a fundamental unit of charge (now known as the elementary charge, e).
  • The elementary charge was found to be approximately[math]1.602 \times 10^{-19} \, \text{C}[/math] .
  • This experiment provided strong evidence for the existence of atoms and the quantization of charge.
  • Impact:
  • – Millikan’s oil drop experiment established the elementary charge as a fundamental constant in physics.
  • – It provided evidence for the atomic structure of matter.
  • – It laid the foundation for quantum mechanics and the development of modern particle physics.
  • The condition for holding a charged oil droplet, of charge Q, stationary between oppositely charged parallel plates is:
  • [math]Q = \frac{mg}{E}[/math]
  • [math]\frac{QV}{d} = mg[/math]
  • Where:
  • – Q is the charge on the oil droplet
  • – m is the mass of the oil droplet
  • – g is the acceleration due to gravity ([math][/math])
  • – E is the electric field strength between the plates
  • Figure 2 Gravitational force and electric force applied on oil drop
  • This is because the electric force (QE) acting on the charged oil droplet must be equal to the weight (mg) of the droplet for it to be held stationary.
  • Derivation:
  • Electric force[math]F_e = QE[/math]
  • Where;
  • [math]E = \frac{V}{d}[/math]
  • Weight[math]F_W = mg[/math]
  • – For equilibrium
  • [math]F_e = F_W[/math]
  • [math]QE = mg[/math]
  • [math]Q = \frac{mg}{E}[/math]
  • [math]QE = mg[/math]
  • [math]\frac{QV}{d} = mg[/math]
  • This equation shows that the charge on the oil droplet (Q) is directly proportional to its mass (m) and inversely proportional to the electric field strength (E).

2. Motion of a falling oil droplet with and without an electric field:

  • Motion of a falling oil droplet:
  • ⇒Without an electric field:
  • – The oil droplet falls under gravity, accelerating downward.
  • – As it falls, air resistance opposes its motion, eventually balancing the force of gravity.
  • – The droplet reaches terminal speed (), where the force of gravity equals the force of air resistance.
  • ⇒With an electric field:
  • – The oil droplet falls under gravity, but the electric field exerts an upward force on the charged droplet.
  • – If the electric field is strong enough, it can balance the force of gravity, suspending the droplet at a fixed point.
  • – If the electric field is weaker, the droplet will still fall, but at a slower rate due to the upward electric force.
  • ⇒Terminal speed:
  • – Without an electric field:
  • [math]v_t = \sqrt{\frac{2mg}{C_d \rho A}}[/math]
  • – With an electric field:
  • [math]v_t = \sqrt{\frac{2(mg – QE)}{C_d \rho A}}[/math]
  • Where:
  • – m is the mass of the droplet
  • – g is the acceleration due to gravity
  • – Q is the charge on the droplet
  • – E is the electric field strength
  • – is the drag coefficient
  • – is the air density
  • – A is the cross-sectional area of the droplet

3. Stokes’ Law for the viscous force on an oil droplet:

  • Stokes’ Law for the viscous force on an oil droplet is:
  • Where:
  • [math]F = 6\pi \eta r v[/math]
  • – F is the viscous force (in Newtons, N)
  • – η (eta) is the dynamic viscosity of the fluid (in[math]\text{Pa} \cdot \text{s}[/math] (Pascal-seconds))
  • – r is the radius of the oil droplet (in meters, m)
  • – v is the velocity of the oil droplet (in meters per second, m/s)
  • This equation describes the force exerted on a small, spherical oil droplet moving through a fluid (like air) due to viscosity. The force opposes the motion of the droplet and is proportional to the droplet’s radius, velocity, and the fluid’s viscosity.
  • – It assumes a laminar flow regime (small Reynolds numbers)
  • – It’s valid for small, spherical particles (like oil droplets)
  • – It’s used to calculate the viscous force on an oil droplet, which can help determine the droplet’s radius, velocity, or viscosity of the surrounding fluid.
  • ⇒Stokes’ Law for the viscous force on an oil droplet used to calculate the droplet radius.
  • The terminal velocity of the droplet can be measured by using a microscope with a calibrated graticule and measuring the distance travelled by the droplet in a certain amount of the time. When the droplet is moving at terminal velocity, the viscous force and weight are equal.
  • [math]Weight(W)=mg[/math]
  • [math]Viscous force(F) = 6\pi \eta r v[/math]
  • Are equal
  • [math]6\pi \eta r v = mg[/math]
  • mass=volume ×density
  • [math]m = \frac{4}{3} \pi r^3 \rho[/math]
  • [math]6\pi \eta r v = \frac{4}{3} \pi r^3 \rho g[/math]
  • [math]r^2 = \frac{9 \eta v}{2 \rho g}[/math]
  • [math]r = \sqrt{\frac{9 \eta v}{2 \rho g}}[/math]
  • Where [math]\rho[/math]  is the density of the oil, r is the droplet’s radius, and  is the viscosity of the fluid.
  • Using the above formula, the radius of the oil droplet can be found meaning that its mass can be measured, and so the charge of the droplet can be calculated.
  • [math]\frac{QV}{d} = mg[/math]
  • [math]\frac{QV}{d} = \frac{4}{3} \pi r^3 \rho g[/math]

4. Significance of Millikan’s results.

  • Millikan’s oil drop experiment (1909) was a groundbreaking study that provided significant results, which:
  • – Determined the elementary charge (e): Millikan measured the charge on individual oil droplets and found that they were all integer multiples of a fundamental unit of charge, now known as the elementary charge (e). This discovery established the quantization of charge.
  • – Confirmed the existence of atoms: Millikan’s results provided strong evidence for the atomic structure of matter, supporting the idea that matter is composed of small, indivisible particles (atoms).
  • – Measured the mass-to-charge ratio (m/e): Millikan’s experiment allowed him to calculate the mass-to-charge ratio for electrons, which was a crucial step in understanding the properties of subatomic particles.
  • – Paved the way for quantum mechanics: Millikan’s discovery of the elementary charge and the quantization of charge laid the foundation for the development of quantum mechanics.
  • – Established the field of particle physics: Millikan’s work contributed to the emergence of particle physics as a distinct field, focusing on the study of subatomic particles and their interactions.
  • – Improved understanding of electricity: Millikan’s results helped clarify the nature of electricity and the behavior of charged particles.
  • – Demonstrated the power of experimentation: Millikan’s oil drop experiment showcased the importance of careful experimentation and measurement in scientific research.

5. Quantization of electric charge.

  • The quantization of electric charge in the oil drop method refers to the fact that the electric charge on the oil droplets was found to be quantized, meaning that it came in discrete packets (quanta) rather than being continuous.
  • Millikan’s experiment showed that the charge on the oil droplets was always an integer multiple of a fundamental unit of charge, which is now known as the elementary charge (e). This means that the charge on the oil droplets was quantized, and the possible values of charge were:
  • Q = ne
  • Where:
  • – Q is the charge on the oil droplet
  • – n is an integer (1, 2, 3, …)
  • – e is the elementary charge (approximately [math]1.6 \times 10^{-19} \, \text{Coulombs}[/math])
  • This quantization of charge was a key discovery in the development of quantum mechanics and the understanding of the behavior of subatomic particles.
  • In the oil drop experiment, the quantization of charge was observed by measuring the charge on individual oil droplets and finding that it was always an integer multiple of the elementary charge. This was a surprising result, as it was expected that the charge would be continuous and could take on any value.
  • The quantization of charge has important implications for our understanding of the behavior of subatomic particles and the structure of matter. It shows that the charge on particles is not continuous, but rather comes in discrete packets, which is a fundamental aspect of quantum mechanics.
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