Animated Solution for Physics - Electrostatics: A charged oil drop is suspended in uniform field of $3\times 10^4{\text{ Vm}}^{-1}$ so that it neither falls nor rises. The charge on the drop will be (take the mass of the charge $=9.9\times 10^{-15}\text{ kg}$ and $g=10{\text{ ms}}^{-2}$)

The magic of levitation isn't just for science fiction; it's a fundamental concept in physics that helped us discover the charge of an electron! Let's dive into the mechanics of a suspended oil drop and see how a simple balance of forces can reveal the invisible.

Imagine a tiny oil drop suspended perfectly still in mid-air. Why isn't it falling? We know that gravity is relentlessly pulling it downwards with a force equal to its weight, $mg$.

For the drop to remain stationary, there must be an equal and opposite force pushing it upwards. In this scenario, that invisible hand is the electric force. Because the drop is placed in a uniform electric field $E$, and it carries a charge $q$, it experiences an upward electric force of magnitude $qE$.

Since the drop neither falls nor rises, it is in a state of static equilibrium. This means the net force acting on it is exactly zero. We can express this delicate balance mathematically:

This elegant equation is the heart of the problem. It tells us that the upward electric pull perfectly cancels out the downward gravitational pull.

Our goal is to find the charge $q$. Let's rearrange our master equation to isolate $q$:

Now, we carefully substitute the values provided in the problem: - Mass of the drop, $m=9.9\times 10^{-15}\text{ kg}$ - Acceleration due to gravity, $g=10{\text{ ms}}^{-2}$ - Electric field strength, $E=3\times 10^4{\text{ Vm}}^{-1}$

Plugging these in, we get:

Let's simplify the numerator first. Multiplying by $10$ changes the power of ten:

Now, we divide the numbers and subtract the exponents:

And there we have it! The charge required to keep this specific oil drop suspended is $3.3\times 10^{-18}\text{ C}$. This simple yet profound balance of forces is the exact principle Robert Millikan used in his famous oil drop experiment to measure the elementary charge of an electron.