JEE Advance - Physics (2016 - Paper 1 Offline - No. 1)
A length-scale (l) depends on the permittivity ($$\varepsilon $$) 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 I is(are) dimensionally correct?
$$l = \sqrt {\left( {{{n{q^2}} \over {\varepsilon {k_b}T}}} \right)} $$
$$l = \sqrt {\left( {{{\varepsilon {k_b}T} \over {n{q^2}}}} \right)} $$
$$l = \sqrt {\left( {{{{q^2}} \over {\varepsilon {n^{2/3}}{k_B}T}}} \right)} $$
$$l = \sqrt {\left( {{{{q^2}} \over {\varepsilon {n^{1/3}}{k_B}T}}} \right)} $$
Explication
$$[n] = [{L^{ - 3}}];[q] = [AT]$$
$$[\varepsilon ] = [{M^{ - 1}}{L^{ - 3}}{A^2}{T^4}]$$
$$[T] = [L]$$
$$[l] = [L]$$
$$[{k_B}] = [{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}]$$
(a) RHS
$$ = \sqrt {{{[{L^{ - 3}}{A^2}{T^2}]} \over {[{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}][{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}][K]}}} $$
$$ = \sqrt {{{[{L^{ - 3}}{A^2}{T^2}]} \over {[{L^{ - 1}}{A^2}{T^2}]}}} = \sqrt {[{L^{ - 2}}]} = [{L^{ - 1}}]$$ Wrong
(b) RHS
$$ = \sqrt {{{[{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}][{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}][K]} \over {[{L^{ - 3}}{A^2}{T^2}]}}} $$
$$ = \sqrt {{{[{L^{ - 1}}{A^2}{T^2}]} \over {[{L^{ - 3}}{A^2}{T^2}]}}} = L$$ Correct
(c) RHS
$$ = \sqrt {{{[{A^2}{T^2}]} \over {[{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}][{L^{ - 2}}][{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}][K]}}} $$
$$ = \sqrt {[{L^3}]} $$ Wrong
(d) RHS
$$ = \sqrt {{{[{A^2}{T^2}]} \over {[{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}][{L^{ - 1}}][{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}][K]}}} $$
$$ = \sqrt {{{[{A^2}{T^2}]} \over {[{L^{ - 2}}{T^2}{A^2}]}}} = [L]$$ Correct
$$[\varepsilon ] = [{M^{ - 1}}{L^{ - 3}}{A^2}{T^4}]$$
$$[T] = [L]$$
$$[l] = [L]$$
$$[{k_B}] = [{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}]$$
(a) RHS
$$ = \sqrt {{{[{L^{ - 3}}{A^2}{T^2}]} \over {[{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}][{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}][K]}}} $$
$$ = \sqrt {{{[{L^{ - 3}}{A^2}{T^2}]} \over {[{L^{ - 1}}{A^2}{T^2}]}}} = \sqrt {[{L^{ - 2}}]} = [{L^{ - 1}}]$$ Wrong
(b) RHS
$$ = \sqrt {{{[{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}][{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}][K]} \over {[{L^{ - 3}}{A^2}{T^2}]}}} $$
$$ = \sqrt {{{[{L^{ - 1}}{A^2}{T^2}]} \over {[{L^{ - 3}}{A^2}{T^2}]}}} = L$$ Correct
(c) RHS
$$ = \sqrt {{{[{A^2}{T^2}]} \over {[{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}][{L^{ - 2}}][{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}][K]}}} $$
$$ = \sqrt {[{L^3}]} $$ Wrong
(d) RHS
$$ = \sqrt {{{[{A^2}{T^2}]} \over {[{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}][{L^{ - 1}}][{M^1}{L^2}{T^{ - 2}}{K^{ - 1}}][K]}}} $$
$$ = \sqrt {{{[{A^2}{T^2}]} \over {[{L^{ - 2}}{T^2}{A^2}]}}} = [L]$$ Correct