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Q.

A length –scale (l) depends on the permittivity (ε)  of a dielectric material ,Boltzmann constant (kB) , 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 expression(s) for l is (are) dimensionally correct?

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a

l=(nq2εkBT)

b

l=(εkBTnq2)

c

l=(q2εn2/3kBT)

d

l=(q2εn1/3kBT)

answer is X.

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Detailed Solution

Permittivity ε=M−1L−3T4I2 Boltzmen constant KB=M1L2T−2K−1Temperature T = KNo.of particles per unit volumen=L−3Charge q=I1T1For A: nq2εKBT=L−3I2T2M−1L−3T4I2M1L2T−2K−1k=1L=L−1For B: εKBTnq2=L            [ from option A]For C: q2εn2/3KBT=I2T2M−1L−3T4I2L−2ML2T−2K−1K=L3/2For D: q2εn1/3KBT=I2T2M−1L−3T4I2L−1ML2T−2K−1k=L
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