Animated Solution for Physics - Gravitation: If $g_E$ and $g_M$ are the accelerations due to gravity on the surfaces of the earth and the moon respectively and if Millikan's oil drop experiment could be performed on the two surfaces, one will find the ratio $\frac{\text{electronic charge on the moon}}{\text{electronic charge on the earth}}$ to be

Millikan's Masterpiece

Imagine you are standing in a laboratory, peering through a microscope at a tiny, suspended droplet of oil. This is the heart of the famous Millikan oil drop experiment. The setup is an elegant dance of forces. Gravity pulls the negatively charged oil drop downwards with a force of $F_g=mg$. To prevent it from falling, an electric field $\vec{E}$ is applied between two parallel plates, pushing the drop upwards with an electric force $F_e=qE$.

When the drop moves through the air, it also experiences a viscous drag force $F_v=6\pi \eta rv$, governed by Stokes' Law. By carefully observing the terminal velocities of the drop as it falls under gravity and rises under the electric field, physicists can calculate the exact charge $q$ residing on the drop.

The question throws a fascinating curveball: What if we pack up this entire apparatus, fly to the moon, and perform the exact same experiment?

We know that the acceleration due to gravity on the moon, $g_M$, is roughly one-sixth of that on Earth, $g_E$. Because gravity is weaker, the downward pull $mg$ on the oil drop will be significantly reduced. Consequently, the terminal velocity of the drop as it falls will be much slower. If you look at the raw data—the velocities and the required balancing electric field—everything will look different on the moon.

Velocities vs. Constants

It is incredibly tempting to look at the formula derived from the experiment, $q=\frac{6\pi \eta r(v_1+v_2)}{E}$, and think, "Wait, the velocities $v_1$ and $v_2$ depend on gravity! If gravity changes, the charge must change!"

While it is absolutely true that the individual velocities $v_1$ and $v_2$ will change because $g$ has changed, the combination of these variables in the final equation perfectly accounts for the new gravitational environment. The experiment is merely a tool to measure something far more profound.

Universality of Charge

The Millikan oil drop experiment measures the electronic charge ($e$). The charge of an electron is an intrinsic, fundamental property of matter. It is a universal constant, approximately $1.6\times 10^{-19}\text{ C}$.

An electron does not care if it is on the Earth, on the Moon, or floating in the void of intergalactic space. Its charge remains absolutely invariant. Therefore, the value of the electronic charge measured on the moon will be exactly the same as the value measured on the earth.

The ratio is simply:

Physics is not just about plugging numbers into formulas; it is about understanding which properties are circumstantial (like weight or velocity) and which properties are fundamental truths of the universe.