Robert Millikan experimented with oil drops to determine a value for the electronic charge - AQA - A-Level Physics - Question 2 - 2022 - Paper 7

To find the mass of the oil droplet, we can use the relationship for terminal velocity:

$mg = \eta \frac{v}{r}$

given that:

• $\eta = 1.8 \times 10^{-5}  \text{N s m}^{-2}$ (viscosity of air)
• $v = 3.8 \times 10^{-5}  \text{m s}^{-1}$ (falling speed)
• The volume of the droplet can be found using its density: $\text{density} = \frac{mass}{volume}$

Calculating the radius using the volume for a sphere, $V = \frac{4}{3} \pi r^3$, gives us the radius. Substituting the appropriate values into the equation yields:

After calculation, the mass m is found to be approximately $8 \times 10^{-16} \, \text{kg}$.

Using the balance of forces on the droplet when it is rising at constant speed:

At rising speed, the electric force $F_e = Q \cdot E$ is acting, where $E = \frac{V}{d}$ is the electric field.

Setting the net force to zero gives:

$F_e - mg = 0$

From this, we can express the relationship:

$F_e = mg \Rightarrow Q \cdot \frac{V}{d} = mg$

Rearranging this, we obtain:

$\frac{Q}{mg} = \frac{d}{V}$

Thus:

$\frac{v_2}{v_1} = \frac{VQ}{dmg} - 1$

Substituting known values from previous parts provides:

• $V = 715 \, V$
• $v_2 = 1.1 \times 10^4 \, \text{m s}^{-1}$
• $d = 0.0116 \, m$

By using these values in the derived equation, the charge Q can be calculated. Comparing this value with the accepted value of the electronic charge ($1.6 \times 10^{-19} \, C$) determines if it is consistent. If discrepancies exist, the reasoning must be retraced to identify potential calculation errors.

Errors in measuring the viscosity of air would influence the calculated terminal velocities and consequently the mass of the oil droplet. A smaller value for viscosity would lead to a smaller calculated mass and charge. This underestimation would result in Millikan's value for the electronic charge appearing larger than its true value, thus impacting the accuracy and reliability of his conclusions about the charge of the electron.