Question-
A slab whose dielectric constant is K, is attached to a string of linear mass density u, while its one end is fixed to a rigid support. Its cross section has sqaure plates of side b and seperation between them is d. Find the fundamental frequency of the vibration of the string.
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Let the length of the plate be ℓ
1
width be b and plate separation be δ. The capacitance, C as a function of x is given by
C(x)=
δ
∈
0
bx
+
δ
k∈
0
b(ℓ−x)
The force on the dielectric is,
F
x
=
2
1
E
2
dx
dC
=−
2
∈
0
(k−1)E
2
δ
b
For the two blocks, we can write,
−T+mg=ma ...(1)
T
2
∈
0
(k−1)E
2
δ
b
=ma ...(2)
∴
dt
2
d
2
x
=a=
2m
1
[mg−
2
∈
0
(k−1)
δ
E
2
b
]=
4
g
or, x=
2
1
(
4
g
)t
2
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