JEE Advanced 2016 · Paper 1 · Physics Q10

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PhysicsMultiple correct+4 / -2Question 10 of 54

A length-scale $(l)$ depends on the permittivity $(\varepsilon)$ 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?

l=\sqrt{\left(\dfrac{nq^2}{\varepsilon k_BT}\right)}

l=\sqrt{\left(\dfrac{\varepsilon k_BT}{nq^2}\right)}

l=\sqrt{\left(\dfrac{q^2}{\varepsilon n^{2/3}k_BT}\right)}

l=\sqrt{\left(\dfrac{q^2}{\varepsilon n^{1/3}k_BT}\right)}

Answer and worked solution

Official answer: B, D

Two energies.

$k_BT$ is an energy.
From Coulomb's law, $\dfrac{q^2}{\varepsilon\,l}$ is an energy, so $\dfrac{q^2}{\varepsilon}=[E][L]$ .

Check each option.

(B) $\dfrac{\varepsilon k_BT}{nq^2}=\dfrac{[E]}{[E][L]\cdot[L^{-3}]}=[L^2]$ . True.
(D) $\dfrac{q^2}{\varepsilon n^{1/3}k_BT}=\dfrac{[E][L]}{[L^{-1}][E]}=[L^2]$ . True.
(A) is the reciprocal of (B), $[L^{-2}]$ . False.
(C) gives $\dfrac{[E][L]}{[L^{-2}][E]}=[L^3]$ . False.

(B), (D)

Topics:Dimensional analysis

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JEE Advanced 2016 · Paper 1, Physics question 10