Stokes' Law (AQA A Level Physics)

Motion of a Falling Oil Droplet

Motion of the Oil Droplet without the Electric Field

• Millikan's experiment was designed to determine the charge of oil droplets, and therefore the charge on electrons

  • To calculate charge using the condition for stationary oil droplets, however, the mass of each oil droplet was required

• With the electric field turned off, the oil droplets fell at terminal velocity

  • This means their weight, mg, was balanced by drag forces as they passed through the air

  • Millikan timed the oil drops as they passed the window of his apparatus to determine their speeds

A diagram showing weight and drag forces on an oil droplet

At terminal velocity, speed is constant and resultant force is zero, so weight is balanced by drag forces

Stokes' Law

Viscous Drag Force using Stokes' Law

• In fluid dynamics, Stokes' law allows drag force on an object through a fluid (such as air) to be calculated, provided:

  • The object is small

  • The object is spherical

  • The speed is low

• Stokes' law states that, for an object with radius r  moving at velocity v  through a fluid with viscosity η, the viscous drag force, F, is:

Calculating Mass from Stokes' Law

• Recall that the drag force and weight are equal at terminal velocity, so:

• Both mass and radius are unknown, but we can write mass in terms of radius for a sphere of known density, ρ :

• Combining these equations allowed Millikan to determine the radius, and therefore mass, of an oil drop whose velocity was known

  • Recall the condition for a stationary oil drop:

• This now gave Millikan enough information to calculate the charge Q  on an oil drop if he knew its terminal velocity and the potential difference V  required to make it stationary

Significance of Millikan's Results

• Millikan repeated his experiment for many different oil droplets

  • They all had different charges, but he noticed an interesting pattern in his results

  • Every value of charge was an integer multiple of 1.60 × 10⁻¹⁹ C

Table of results from Millikan's Experiment

Millikan's results showed that the charge on each oil drop was an integer multiple of 1.60 × 10⁻¹⁹ C

Quantisation of Electric Charge

• Each negative oil droplet had been ionised, so carried an integer number of electrons

  • The fact that each droplet's charge was quantised into integer multiples of 1.60 × 10⁻¹⁹ C suggested that this was the charge of each electron