Stokes' Law (AQA A Level Physics): Revision Note

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

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 $\eta$, the viscous drag force, $F$, is:

$F=6\mathrm{\pi }\eta rv$

Calculating mass from Stokes' law

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

$6\mathrm{\pi }\eta rv=mg$

• Both mass and radius are unknown, but mass can be written in terms of radius for a sphere of known density, $\rho$:

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

• 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:

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

• 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

• 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