F = 6πηrv Significance of Millikan’s results.

The idea

Stokes’ law gives the viscous drag on a small sphere moving slowly through a fluid as F = 6πηrv, where η is the fluid viscosity, r the sphere radius and v its speed. In Millikan’s experiment, charged oil droplets fall under gravity and are held or moved by an electric field between horizontal plates. When a droplet falls at terminal velocity, the drag balances the net downward force, allowing r and then the droplet charge to be found. Repeating measurements showed every droplet charge was an integer multiple of a smallest value, the elementary charge e. This demonstrated quantisation of electric charge and gave a value for e, supporting the idea that charge exists in discrete units carried by electrons.

What earns the marks

• Stokes’ law F = 6πηrv applies to a small sphere moving slowly through a viscous fluid, where η is the viscosity of the fluid, r is the sphere radius and v is the sphere speed.
• At terminal velocity the resultant force on a falling droplet is zero, so viscous drag balances the net downward force, which is weight minus upthrust.
• Measuring terminal velocity and using F = 6πηrv allows the droplet radius r to be determined, because η is known for the fluid and v is measured.
• With the radius known, the weight and upthrust can be calculated, and applying an electric field lets the charge on the droplet be found from the balance of electric force and other forces.
• Millikan’s repeated results showed droplet charges were whole-number multiples of a smallest charge, demonstrating quantisation of electric charge.
• The smallest common charge gave a value for the elementary charge e, now known as the magnitude of the charge on an electron.
• The experiment provided evidence that electric charge is carried in discrete units rather than being continuous.

Common mistakes and how to put them right

• Treating F = 6πηrv as the weight of the droplet.

  Put it right by stating that it is the viscous drag force, while weight is mg and upthrust is the weight of displaced fluid.

• Using the droplet’s instantaneous velocity instead of its terminal velocity.

  Put it right by measuring the steady speed reached when the resultant force is zero.

• Assuming every droplet carries exactly one elementary charge.

  Put it right by explaining that droplets carry integer multiples of e, and the smallest common value is identified from many measurements.

Exam tips

• When deriving or using F = 6πηrv, state clearly that it applies to a small sphere moving slowly through a viscous fluid and define each symbol.
• In calculations, first find terminal velocity from distance and time, then use F = 6πηrv to find r before attempting to find charge.
• When explaining significance, link the numerical results to quantisation and to a value for e rather than only describing the apparatus.