12.1.4 Principle of Millikan's determination of the electronic charge, e

Apparatus and Method

• Atomiser: Produces tiny oil droplets that carry a negative charge.
• Electric Field: When a potential difference ($V$) is applied across the plates, a uniform electric field is created. Each charged oil droplet experiences an electric force in the direction of the field.
• Microscope: Used to observe the droplet and measure its behaviour within the field. The electric field strength (E) can be expressed as:

$$E = \frac{V}{d}$$

where:

• $V$ is the potential difference across the plates,
• $d$ is the separation between the plates. When an oil droplet becomes stationary, its weight is balanced by the electric force acting on it:

$$EQ = mg$$

Thus:

$$\frac{QV}{d} = mg$$

where:

• $Q$ is the charge on the droplet,
• $m$ is the mass of the droplet,
• $g$ is the acceleration due to gravity.

Finding the Mass of the Oil Droplet

To determine the charge on the droplet, we need to know its mass, which requires additional steps:

1. Remove the Electric Field: When the potential difference is turned off, the droplet begins to fall due to gravity.
2. Viscous Drag: As the droplet falls, it reaches terminal velocity (where gravitational force is balanced by the viscous drag force). The viscous drag force $F$ on the droplet, according to Stokes' Law, is:

$$F = 6 \pi \eta r v$$

where:

• $\eta$ is the viscosity of the air,
• $r$ is the radius of the oil droplet,
• $v$ is the terminal velocity. At terminal velocity:

$$6 \pi \eta r v = mg$$

Since the droplet's mass is related to its volume $( \text{volume} = \frac{4}{3} \pi r^3 )$ and density $\rho$, we can rewrite:

$$m = \frac{4}{3} \pi r^3 \rho$$

Substituting this back into the drag equation and solving for $r$:

$$r = \sqrt{\frac{9 \eta v}{2 \rho g}}$$

With $r$ known, $m$ can be calculated, allowing the charge $Q$ to be determined from:

$$\frac{QV}{d} = mg$$

Key Findings and Significance

Millikan's measurements showed that the charges on the oil droplets were always multiples of 1.6 × 10⁻¹⁹ C. This supported the idea that charge is quantised, meaning it exists in discrete "packets" or units. The smallest observed charge was interpreted as the charge of a single electron.

Millikan's experiment was significant as it demonstrated that:

• Electric charge is quantised, existing in multiples of a fundamental unit,
• The fundamental charge, $e$, is approximately 1.6 × 10⁻¹⁹ C,
• This provided critical support for atomic theory and the understanding of electrons as discrete particles carrying a fixed charge.