Animated Solution for Physics - Electrostatics: Comprehension Passage

Imagine this cylindrical chamber. The bottom plate is at a high positive potential, $+V_0$, and the top plate is at a negative potential, $-V_0$. The little conducting balls resting at the bottom will acquire a positive charge, and because like charges repel, they will be shot upwards towards the top plate!

Let's focus on just one ball. Since it's touching the bottom plate, its potential becomes $V_0$. We know the capacitance of a small sphere is $4\pi {\epsilon }_{0}r$. So, the charge $q$ it picks up is simply its capacitance times the potential, which gives us:

Now, what's the electric field driving this ball? The chamber acts like a giant parallel plate capacitor. The total potential difference is $V_0-(-V_0)$, which is $2V_0$. Dividing this by the height $h$ gives us the uniform electric field $E$ inside the chamber:

With the charge and electric field known, we can find the electrostatic force acting on the ball. The force $F$ is simply the charge $q$ times the electric field $E$. This upward force is what propels the ball towards the top plate.

By Newton's second law, the acceleration $a$ is this force divided by the mass $m$. Substituting our expressions for $q$ and $E$, we get:

Notice carefully that the acceleration is directly proportional to the square of $V_0$.

The ball starts from rest and travels a distance $h$. Using the second equation of motion, $h=\frac{1}{2}a{t}^2$, we can solve for the time of flight, $t$:

When we plug in our acceleration, notice that $V_0^2$ is in the denominator inside the square root. So, the time $t$ is inversely proportional to $V_0$.

Finally, let's find the average current. Current is the total charge transported per unit time. For $n$ balls, it's $n$ times $q$ divided by $t$:

Since the charge $q$ is proportional to $V_0$, and the time $t$ is inversely proportional to $V_0$, dividing them gives us a current that is proportional to $V_0^2$! A beautiful result.

Let's also answer the conceptual question. When a ball hits the top plate, the coefficient of restitution is zero, meaning it stops momentarily. However, it now touches the negative plate and acquires a negative charge!

The upward electric field now exerts a downward force on this negative charge, pushing it back to the bottom plate. So, it bounces back carrying the opposite charge. And since the force is constant, not proportional to displacement, it's an oscillatory motion, but not Simple Harmonic Motion.