JEE Advanced · 2016 · 22 May 2016 · Paper 1 · Units & Measurement

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 expression(s) for

l

is(are) dimensionally correct?

Answer: (B), (D)

Solution

Dimensional analysis:

Option (A):

\frac{n q ^{2}}{ε k _{B} T} \sim \frac{[ L ^{- 3} ] [ Q ^{2} ]}{[ Q ^{2} / ( F L ^{2} )] [ F L ]} = [L^{- 2}]

, so

\sim [L^{- 1}]

. Incorrect.

Option (B):

\frac{ε k _{B} T}{n q ^{2}} \sim \frac{[ Q ^{2} / ( F L ^{2} )] [ F L ]}{[ L ^{- 3} ] [ Q ^{2} ]} = [L^{2}]

, so

\sim [L]

. Correct.

Option (C):

\frac{q ^{2}}{ε n ^{2/3} k _{B} T} \sim \frac{[ Q ^{2} ]}{[ Q ^{2} / ( F L ^{2} )] [ L ^{- 2} ] [ F L ]} = [L^{- 1}]

, so

\sim [L^{- 1/2}]

. Incorrect.

Option (D):

\frac{q ^{2}}{ε n ^{1/3} k _{B} T} \sim \frac{[ Q ^{2} ]}{[ Q ^{2} / ( F L ^{2} )] [ L ^{- 1} ] [ F L ]} = [L^{2}]

, so

\sim [L]

. Correct.

JEE Advanced · 2016 · 22 May 2016 · Paper 1 · One or more correct options