Electrostatics> In fig, the capacitors have plate area. A...
1 AnswersTo determine the time period of oscillation for a dielectric slab displaced slightly in a capacitor, we first need to understand the forces acting on the slab and how they relate to simple harmonic motion (SHM). Let's break this down step by step.
We have a parallel plate capacitor with a dielectric slab of mass m inserted between the plates. The area of the plates is given by A = l × b, where l is the length and b is the breadth. The separation between the plates is d.
When the slab is displaced slightly from its equilibrium position, it experiences a restoring force due to the electric field between the capacitor plates. This electric field E can be expressed as:
The force F acting on the dielectric slab can be derived from the energy stored in the capacitor. The energy U stored in a capacitor is given by:
For a parallel plate capacitor, the capacitance C is:
When the slab is displaced by a small distance x, the capacitance changes, and thus the energy changes, leading to a restoring force. The change in capacitance ΔC can be approximated for small displacements:
The change in energy ΔU can be expressed as:
The restoring force can then be derived from the change in energy with respect to displacement:
For the system to exhibit simple harmonic motion, the restoring force must be proportional to the displacement, which can be expressed as:
From the force derived above, we can find that the effective spring constant k is related to the change in capacitance and the electric field. The relationship can be established as:
The time period T of a mass-spring system undergoing SHM is given by:
Substituting our expression for k into this formula will yield the time period of oscillation for the dielectric slab. The exact expression will depend on the parameters of the system, including the voltage and the dimensions of the capacitor.
In summary, yes, the oscillation of the dielectric slab can be classified as simple harmonic motion, provided the displacement is small enough for the linear approximation to hold. The time period can be calculated using the derived expressions for the effective spring constant and the mass of the slab. This analysis highlights the interplay between electric forces and mechanical motion in capacitive systems.
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