A length--scale depends on the permittivity of a dielectric material, Boltzmann constant , the absolute temperature , the number per unit volume of certain charged particles, and the charge carried by each of the particles. Which of the following expr

Question

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? [2016]

Smart Achievers · Accepted Answer

(2, 4)

$Dimensions of [n] = [L^{- 3}], [q] = [A T]$

$[\frac{q^{2}}{ε}] = [F r^{2}] = [U \times r] = [M L^{3} T^{- 2}]$

$[k_{B} T] = [U] = [M L^{2} T^{- 2}]$

$Dimension of l = [L]$

$(a) R.H.S. = \sqrt{(\frac{q^{2}}{ε}) \times \frac{1}{(k_{B} T)} \times n}$
$= \sqrt{[U \times r] \times \frac{1}{[U]} \times n} = \sqrt{[n] [r]} = \sqrt{[L^{- 3}] [L]} = [L^{- 1}]$

$(b) R.H.S. = \sqrt{\frac{ε (k_{B} T)}{n q^{2}}} = \sqrt{\frac{(k_{B} T)}{n (q^{2} / ε)}}$

$= \sqrt{\frac{[U]}{[n] [U \times r]}} = \sqrt{\frac{1}{[n] [r]}} = \sqrt{\frac{1}{[L^{- 3}] [L]}} = [L]$

$(c) R.H.S. = \sqrt{(\frac{q^{2}}{ε}) \times \frac{1}{(k_{B} T)} \times \frac{1}{n^{2 / 3}}}$

$= \sqrt{[U \times r] \times \frac{1}{[U]} \times \frac{1}{[L^{- 2}]}} = [L^{3 / 2}]$

$(d) R.H.S. = \sqrt{(\frac{q^{2}}{ε}) \times \frac{1}{(k_{B} T)} \times \frac{1}{n^{1 / 3}}}$

$= \sqrt{[U \times r] \times \frac{1}{[U]} \times \frac{1}{[n^{1 / 3}]}} = \sqrt{[r] \times \frac{1}{[n^{1 / 3}]}}$

$= \sqrt{[L] \frac{1}{[L^{- 1}]}} = [L]$

Solution

1 $l = \sqrt{(\frac{n q^{2}}{ε k_{B} T})}$

2 $l = \sqrt{(\frac{ε k_{B} T}{n q^{2}})}$

3 $l = \sqrt{(\frac{q^{2}}{ε n^{2 / 3} k_{B} T})}$

4 $l = \sqrt{(\frac{q^{2}}{ε n^{1 / 3} k_{B} T})}$

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Solution

1 l=(nq2εkBT)

2 l=(εkBTnq2)

3 l=(q2εn2/3kBT)

4 l=(q2εn1/3kBT)

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1 $l = \sqrt{(\frac{n q^{2}}{ε k_{B} T})}$

2 $l = \sqrt{(\frac{ε k_{B} T}{n q^{2}})}$

3 $l = \sqrt{(\frac{q^{2}}{ε n^{2 / 3} k_{B} T})}$

4 $l = \sqrt{(\frac{q^{2}}{ε n^{1 / 3} k_{B} T})}$