
A dielectric slab of dielectric constant , mass , thickness , and area is hanging vertically in equilibrium under the influence of gravity and electrostatic pull of a capacitor connected to a battery of voltage . The capacitor has plates of area and distance between plates is . The capacitor is half-filled by the dielectric. Suddenly a mass of is attached to the dielectric without any impulse on the system. The slab falls off in time . Evaluate (in a sec). Take , , , .

Answer
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Hint: We will first evaluate the new capacitance after the dielectric is inserted in it. Then using that capacitance we will evaluate the electrostatic pull due to the battery and using these parameters and given quantities we will evaluate the time the slab falls off when some mass is introduced without any impulse.
Formula used:
The formula of the capacitance
Electrostatic potential energy formula
Electrostatic force
.
Complete step by step solution:
Suppose at any point , the length of the dielectric is inserted between the capacitor .
Hence the new capacitance will be given as
where is the dielectric constant and is the length of the dielectric introduced.
……….
Now using this dielectric we will evaluate the potential energy which can be given by , hence
Substituting the value of capacitance from the equation , we get
……….
We know that the electrostatic pull is acting due to battery of voltage which can be given as
Hence differentiating the equation with respect to what we get,
Also given that this electrostatic force equals to gravity acting on the dielectric which results as
Hence on comparing it
………
Now the Net force can be given as by the difference between the initial condition of the dielectric and its final condition ,
Hence the acceleration can be given as
………
Form equation we can conclude the value of mass as
Substituting it in the equation , we get
Now the slab is half-filled with dielectric hence the height of the dielectric will be given as
Now evaluating the time by the formula for height and
Hence the time the slab falls off when some mass is introduced without any impulse is .
Note:
Dielectric is a material that is insulating. In other words it is a very poor conductor of electricity. When we place dielectrics in the electric fields, then current flows through them practically as they don’t have free electrons and loose bounds.
Formula used:
The formula of the capacitance
Electrostatic potential energy formula
Electrostatic force
Complete step by step solution:
Suppose at any point
Hence the new capacitance will be given as
where
Now using this dielectric we will evaluate the potential energy which can be given by
Substituting the value of capacitance from the equation
We know that the electrostatic pull is acting due to battery of voltage
Hence differentiating the equation
Also given that this electrostatic force equals to gravity acting on the dielectric which results as
Hence on comparing it
Now the Net force
Hence the acceleration can be given as
Form equation
Substituting it in the equation
Now the slab is half-filled with dielectric hence the height of the dielectric
Now evaluating the time by the formula for
Hence the time the slab falls off when some mass is introduced without any impulse is
Note:
Dielectric is a material that is insulating. In other words it is a very poor conductor of electricity. When we place dielectrics in the electric fields, then current flows through them practically as they don’t have free electrons and loose bounds.
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