Question

Plates of the capacitor have plate area A and are clamped in the laboratory. The dielectric slab is released from rest with a length a inside the capacitor. Neglecting any effect of friction or gravity, show that the slab will execute periodic motion and find its time period.

Answer 1

The provided problem statement describes a scenario that leads to a constant force on the dielectric slab, resulting in uniform acceleration, not periodic motion. Therefore, it is not possible to show that the slab will execute periodic motion based on the given information and standard physics principles.

Answer 2

The force on the dielectric slab can be determined by considering the change in potential energy stored in the capacitor. Since the capacitor is connected to a battery, the voltage across it remains constant, denoted by $\mathcal{E}$. The energy stored in the capacitor is given by $U = \frac{1}{2} C \mathcal{E}^2$. The force acting on the slab is the negative gradient of this potential energy with respect to the position of the slab: $F = -\frac{dU}{dx}$.

Let 'x' be the length of the dielectric slab inside the capacitor. The capacitance $C(x)$ depends on 'x'. Assume the dielectric slab has length 'a' and width W. The capacitor plates have separation 'd' and plate area A. Let the length of the capacitor plates be $L_{cap}$, so $A = L_{cap}W$.

When a length 'x' of the dielectric is inside the capacitor, the total capacitance can be viewed as two capacitors in parallel:

1. The portion filled with the dielectric: length 'x', width W. Its capacitance is $C_1 = \frac{K \epsilon_0 xW}{d}$.
2. The portion filled with air: length $L_{cap}-x$, width W. Its capacitance is $C_2 = \frac{\epsilon_0 (L_{cap}-x)W}{d}$.

The total capacitance is $C(x) = C_1 + C_2 = \frac{K \epsilon_0 xW}{d} + \frac{\epsilon_0 (L_{cap}-x)W}{d} = \frac{\epsilon_0 W}{d} (Kx + L_{cap}-x) = \frac{\epsilon_0 W}{d} (L_{cap} + (K-1)x)$. Since $A = L_{cap}W$, we can write $C(x) = \frac{\epsilon_0}{d} (A + (K-1)xW)$.

The potential energy stored in the capacitor is $U(x) = \frac{1}{2} C(x) \mathcal{E}^2 = \frac{1}{2} \frac{\epsilon_0}{d} (A + (K-1)xW) \mathcal{E}^2$.

The force on the dielectric slab is $F(x) = -\frac{dU}{dx} = -\frac{1}{2} \frac{\epsilon_0 W}{d} (K-1) \mathcal{E}^2$.

This force is constant and independent of 'x' as long as the dielectric is partially inside the capacitor ($0 < x < L_{cap}$). If $K>1$, the force is attractive, pulling the dielectric into the capacitor.

The problem states that the slab will execute periodic motion. This implies that the force must be a restoring force, meaning it is directed towards an equilibrium position and is proportional to the displacement from that equilibrium. A constant force would lead to uniform acceleration, not periodic motion.

Let's re-examine the setup. The dielectric slab has length 'a'. It is released from rest with length 'a' inside the capacitor. This implies that the capacitor plates must have a length $L_{cap}$ that is greater than or equal to 'a'.

If the capacitor plates are sufficiently long ($L_{cap} \ge a$), and the dielectric is released from rest, the force calculated above, $F = -\frac{1}{2} \frac{\epsilon_0 W}{d} (K-1) \mathcal{E}^2$, is a constant force. This force would cause the dielectric to accelerate uniformly. This contradicts the premise of periodic motion.

There might be a misunderstanding of the problem setup or a missing detail. However, if we are to show that the motion is periodic, we must assume a scenario where a restoring force exists.

A common scenario for periodic motion in such a system is when the dielectric is pulled into the capacitor, and then some other force (e.g., a spring, or the dielectric being pushed out by some mechanism) causes it to oscillate. Without such a mechanism, a constant attractive force would simply pull the dielectric in until it is fully inside, and then it would remain there (or move with constant velocity if it were already fully inside and the force was still acting).

Let's consider the possibility that the problem implies that the dielectric is inserted such that it experiences a force that varies with position in a way that leads to oscillation. This typically happens when the dielectric is partially inserted and then pulled in, and then some other mechanism provides the restoring force.

However, if we assume the problem statement is complete and the motion is indeed periodic, there might be an interpretation where the force itself, derived from the potential, becomes a restoring force under specific conditions not immediately obvious from the text.

Let's assume the problem implies that the dielectric is pulled into the capacitor and then, due to its inertia, overshoots an equilibrium point and oscillates. If the capacitor plates have a finite length, $L_{cap}$, and the dielectric has length 'a', the force $F(x) = -\frac{1}{2} \frac{\epsilon_0 W}{d} (K-1) \mathcal{E}^2$ is constant for $0 < x < L_{cap}$.

If the dielectric is released from rest at $x=0$ (just starting to enter), it will accelerate with constant acceleration $a = \frac{F}{m} = \frac{1}{m} \frac{1}{2} \frac{\epsilon_0 W (K-1) \mathcal{E}^2}{d}$. This is not periodic motion.

Perhaps the question implies that the dielectric is inserted to some point, and then released, and the restoring force that causes oscillation is implicitly assumed. However, based on the provided information, the force calculated is constant.

Let's consider the possibility that the problem statement is flawed or incomplete as presented, as a constant force does not lead to periodic motion. If we must show periodic motion, we would need a force that is a function of displacement and is directed towards an equilibrium position, e.g., $F(x) = -kx$.

If we assume that the dielectric is inserted into a region where the capacitance changes in a way that produces a restoring force, we would need a different capacitance formula.

Given the problem statement, it's impossible to rigorously derive periodic motion from the provided information because the calculated force is constant. The problem might be intended to be illustrative or assumes a context not provided.

However, if we assume that the setup leads to simple harmonic motion, then the force must be of the form $F = -kx$. The potential energy would then be $U(x) = \frac{1}{2}kx^2$. The kinetic energy is $T = \frac{1}{2}mv^2$. For simple harmonic motion, the period is $T = 2\pi\sqrt{\frac{m}{k}}$.

If we were to force a periodic motion interpretation from the constant force $F_0 = \frac{1}{2} \frac{\epsilon_0 W (K-1) \mathcal{E}^2}{d}$, this force would cause acceleration $a = F_0/m$. If this acceleration were somehow to reverse direction periodically, it would imply an oscillating potential, which isn't present here.

Let's consider a scenario where the dielectric is inserted to a length $x_0$ and then released. If $x_0$ is the midpoint of some oscillation, the force should be zero there. But the force is constant and non-zero.

Conclusion based on standard physics principles: With a constant voltage source and a dielectric whose insertion leads to a linearly changing capacitance (as derived), the force on the dielectric is constant. A constant force causes uniform acceleration, not periodic motion. The problem statement as given cannot be solved to show periodic motion.

If this were a multiple-choice question and one of the options was "The premise is incorrect, the motion is not periodic," that would be the correct answer.

Assuming there is a context missing or an intended interpretation that leads to periodic motion, we cannot proceed to find the period without a valid restoring force expression.

Let's assume, for the sake of providing an answer, that the problem setter intended a scenario where the force is proportional to displacement from some equilibrium position. However, the derivation above shows a constant force. Therefore, I cannot fulfill the request to "show that the slab will execute periodic motion" based on the provided text and standard physics principles.

If we hypothetically assume that the force is $F(x) = -kx$, then the motion is periodic with period $T = 2\pi\sqrt{\frac{m}{k}}$. But we cannot derive $k$ from the given information to show periodic motion.