Advertisements
Advertisements
प्रश्न
Consider the situation shown in figure. The 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.
उत्तर
Given that the area of the plates of the capacitors is A.
As ''a'' length of the dielecric slab is inside the capacitor.
Therefore, the area of the plate covered with dielectric is `= A/la`
The capacitance of the portion with dielectric is given by `C_1 = (K∈_0Aa)/(ld)`
The capacitance of the portion without dielectric is given by `C_1 = (∈_0A(l-a))/(ld)`
The two parts can be considered to be in parallel.
Therefore, the net capacitance is given by
`C = C_1 + C_2`
`⇒ C = (∈_0A)/(ld)[Ka + (l-a)]`
`⇒ C = (∈_0A)/(ld)[l + a(K-1)]`
Let us consider a small displacement da of the slab in the inward direction. The capacitance will increase, therefore the energy of the capacitor will also increase. In order to maintain constant voltage, the battery will supply extra charges, therefore the battery will do work.
Work done by the battery = change in energy of capacitor + work done by the force F on the capacitor
`dW_B = dU + dW_F`
Let the charge dq is supplied by the battery, and the change in the capacitor be dC
`dW_B = (dq).V = (dC).V^2`
`dU = 1/2(dC).V^2`
`(dC).V^2 = 1/2(dC).V^2 + F.da`
`1/2(dC).V^2 = F.da`
`⇒ F = 1/2 (dC)/(da)V^2`
`⇒ F = 1/2 d/(da) ((∈_0A)/(ld)[l + a(K-1)])V^2`
`⇒ F = 1/2 (∈_0A)/(ld) (K - 1)`
The acceleration of the dielectric is given by `a_0 = 1/2 (∈_0A)/(ldm)(K-1)`
As, the force is in inward direction, it will tend to make the dielectric to completely fill the space inside the capacitors. As, the dielectric completely fills the space inside the capacitor at this instant its velocity is not zero. The dielectric slab tends to move outside the capacitor. As the slab tends to move out, the direction of the force due to the capacitor will reverse its direction. Thus, the dielectric slab will have a periodic motion.
The time taken to move distance `(l-a)` can be calculated as :-
`(l - a) = 1/2 a_0t^2`
`t = sqrt ((2(l-a))/(a_0))`
`t = sqrt (2(l-a) xx (2ldm)/(∈_0AV^2(K-1))`
`t = sqrt ((4m(l-a)ld)/(∈_0AV^2(K-1))`
For the complete cycle the time period will be four times the time taken for covering distance `(l-a)`.
It is given by :-
`T = 4t = 4 xx 2 sqrt((m(l-a)ld)/(∈_0AV^2(K-1))) = 8 sqrt ((m(l-a)ld)/(∈_0AV^2(K-1))`
APPEARS IN
संबंधित प्रश्न
A capacitor of capacitance C is charged fully by connecting it to a battery of emf E. It is then disconnected from the battery. If the separation between the plates of the capacitor is now doubled, how will the following change?
(i) charge stored by the capacitor.
(ii) Field strength between the plates.
(iii) Energy stored by the capacitor.
Justify your answer in each case.
A capacitor has capacitance C. Is this information sufficient to know what maximum charge the capacitor can contain? If yes, what is this charges? If no, what other information is needed?
Three capacitors of capacitances 6 µF each are available. The minimum and maximum capacitances, which may be obtained are
Following operations can be performed on a capacitor:
X − connect the capacitor to a battery of emf ε.
Y − disconnect the battery.
Z − reconnect the battery with polarity reversed.
W − insert a dielectric slab in the capacitor.
(a) In XYZ (perform X, then Y, then Z) the stored electric energy remains unchanged and no thermal energy is developed.
(b) The charge appearing on the capacitor is greater after the action XWY than after the action XYZ.
(c) The electric energy stored in the capacitor is greater after the action WXY than after the action XYW.
(d) The electric field in the capacitor after the action XW is the same as that after WX.
Find the charge appearing on each of the three capacitors shown in figure .
It is required to construct a 10 µF capacitor which can be connected across a 200 V battery. Capacitors of capacitance 10 µF are available but they can withstand only 50 V. Design a combination which can yield the desired result.
A capacitor is made of a flat plate of area A and a second plate having a stair-like structure as shown in figure . The width of each stair is a and the height is b. Find the capacitance of the assembly.
Find the equivalent capacitance of the infinite ladder shown in figure between the points A and B.
A 5⋅0 µF capacitor is charged to 12 V. The positive plate of this capacitor is now connected to the negative terminal of a 12 V battery and vice versa. Calculate the heat developed in the connecting wires.
A parallel-plate capacitor of plate area A and plate separation d is charged to a potential difference V and then the battery is disconnected. A slab of dielectric constant K is then inserted between the plates of the capacitor so as to fill the space between the plates. Find the work done on the system in the process of inserting the slab.
An air-filled parallel-plate capacitor is to be constructed which can store 12 µC of charge when operated at 1200 V. What can be the minimum plate area of the capacitor? The dielectric strength of air is `3 xx 10^6 "Vm"^-1`
A parallel-plate capacitor with the plate area 100 cm2 and the separation between the plates 1⋅0 cm is connected across a battery of emf 24 volts. Find the force of attraction between the plates.
Three capacitors C1 = 3μF, C2 = 6μF, and C3 = 10μF are connected to a 50 V battery as shown in Figure below:
Calculate:
(i) The equivalent capacitance of the circuit between points A and B.
(ii) The charge on C1.
Three circuits, each consisting of a switch 'S' and two capacitors, are initially charged, as shown in the figure. After the switch has been closed, in which circuit will the charge on the left-hand capacitor
(i) increase,
(ii) decrease, and
(iii) remains the same? Give reasons.
Dielectric constant for a metal is ______.
A capacitor of capacity 2 µF is charged to a potential difference of 12 V. It is then connected across an inductor of inductance 0.6 mH. The current in the circuit at a time when the potential difference across the capacitor is 6.0 V is ______ × 10-1A.
