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JEE Advanced 2016 Paper 1
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© examsnet.com
Question : 10
Total: 54
A length-scale
(
l
)
depends on the permittivity
(
ε
)
of a dielectric material, Boltzmann constant
k
B
, the absolute temperature T, the number per unit volume (n) of certain charged particles, and the charge (q) carried by each of the particles, Which of the following expressions(s) for
l
is(are) dimensionally correct?
l
=
√
(
n
q
2
ε
k
B
T
)
l
=
√
(
ε
k
B
T
n
q
2
)
l
=
√
(
q
2
ε
n
2
3
k
B
T
)
l
=
√
(
q
2
ε
n
1
3
k
B
T
)
Validate
Solution:
We know,
F
=
1
4
π
ε
0
q
2
r
2
⇒
q
2
ε
0
=
(
F
r
2
)
4
π
So,dimension
q
2
ε
0
=
d
i
m
(
F
r
2
)
=
M
L
T
−
2
×
L
2
=
M
L
3
T
2
Similarly E
=
3
2
K
B
T
⇒
d
i
m
(
K
B
T
)
=
dim(Energy)
=
M
L
2
T
−
2
(A)
√
n
q
2
ε
k
B
T
=
√
L
−
3
×
M
L
3
T
−
2
M
L
2
T
−
2
=
1
L
(B)
√
(
E
)
×
v
o
l
F
r
2
=
√
M
L
2
T
−
2
×
L
3
M
L
T
−
2
×
L
2
=
L
(C)
√
F
r
2
(
v
o
l
)
2
3
(
K
ε
)
=
√
M
L
T
−
2
×
L
2
×
L
2
M
L
2
T
−
2
=
√
L
3
=
L
3
2
(D)
√
F
r
2
(
v
o
l
)
1
3
E
n
e
r
g
y
=
√
M
L
T
−
2
L
2
×
L
M
L
2
T
−
2
=
L
[∴dimension n = dim
(
1
v
o
l
)
=
L
−
3
]
© examsnet.com
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