QV = mg d Motion of a falling oil droplet with and without an electric…

The idea

In Millikan’s experiment, a charged oil droplet falls under gravity and is then held by an electric field. Without the field, the droplet reaches terminal speed when its weight is balanced by viscous drag (upthrust is considered negligible). Measuring that speed gives the radius and mass using Stokes' Law. With a vertical electric field, the droplet can be held stationary. When the electric force balances the weight, $QV/d = mg$, where $Q$ is the droplet charge, $V$ is the potential difference, $m$ is the mass, $g$ is gravitational field strength and $d$ is the plate separation. Combining the measurements allows $Q$ to be determined. Repeating for many droplets shows that charge is quantised in multiples of $e$.

What earns the marks

• Without the electric field, the droplet falls at terminal speed when the resultant force is zero: weight is balanced by viscous drag (ignoring upthrust).
• The terminal speed without the field is used with Stokes’ Law to find the droplet radius, and hence its volume and mass.
• With the electric field applied, the electric force on the droplet is $QV/d$, where $V$ is the potential difference across plates separated by $d$.
• When the droplet is held stationary, the electric force balances the weight: $QV/d = mg$, which rearranges to $QV = mgd$.
• The mass $m$ is obtained from the radius and the density of the oil, so $Q$ can be calculated from $Q = mgd/V$.
• Repeating the measurement for many droplets gives values of $Q$ that are integer multiples of the elementary charge $e$, demonstrating charge quantisation.

Common mistakes and how to put them right

• Including upthrust when balancing forces on the falling droplet.

  Put it right: Ignore upthrust as it is negligible for oil droplets in air, so weight equals viscous drag.

• Using the droplet diameter instead of its radius in Stokes’ Law.

  Put it right: Calculate the radius from the terminal speed and then use that radius consistently.

• Confusing the distance between the plates $d$ with the distance the droplet falls.

  Put it right: Ensure $d$ in $V/d$ is the plate separation.

Exam tips

• Sketch the forces on the droplet for both cases: falling without the field and held stationary with the field. Label weight, viscous drag, and electric force.
• Write the force balance equation before substituting numbers, and check that each symbol has its correct SI unit.
• When using $QV/d = mg$, rearrange algebraically for the required quantity before inserting values to reduce arithmetic slips.