JEE MAIN - Physics (2018 - 15th April Evening Slot - No. 22)
The characteristic distance at which quantum gravitational effects are significant, the Planck length, can be determined from a suitable combination of the fundamental physical constants G, h and c.
Which of the following correctly gives the Planck length ?
Which of the following correctly gives the Planck length ?
G $$\hbar $$2 c3
G2 $$\hbar $$ c
$${G^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}}}{\hbar ^2}c$$
$${\left( {{{G\hbar } \over {{c^3}}}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}}}$$
Explanation
Plank length,
$$\ell $$ = k Gp $$\hbar $$q Cr
[ Mo L To] = [ M$$-$$1 L3 T$$-$$2 ]p [ M L2 T$$-$$1] q [ L T$$-$$1]r
[Mo L To ] = [M$$-$$p + q L(3p + 2q + r) T$$-$$(2p + q + r)]
Comparing both sides,
$$-$$ p + q = 0
3p + 2q + r = 1
$$-$$ (2p + q + r) = 0
Solving those equation we get,
p = $${1 \over 2},$$ q = $${1 \over 2},$$ $$r = - {3 \over 2}$$
$$\therefore\,\,\,$$ $$\ell $$ = k G$${^{{1 \over 2}}}$$ $${\hbar ^{{1 \over 2}}}$$ $${C^{ - {3 \over 2}}}$$
= $${\left( {{{G\hbar } \over {{C^3}}}} \right)^{{1 \over 2}}}$$
(assume k = 1)
$$\ell $$ = k Gp $$\hbar $$q Cr
[ Mo L To] = [ M$$-$$1 L3 T$$-$$2 ]p [ M L2 T$$-$$1] q [ L T$$-$$1]r
[Mo L To ] = [M$$-$$p + q L(3p + 2q + r) T$$-$$(2p + q + r)]
Comparing both sides,
$$-$$ p + q = 0
3p + 2q + r = 1
$$-$$ (2p + q + r) = 0
Solving those equation we get,
p = $${1 \over 2},$$ q = $${1 \over 2},$$ $$r = - {3 \over 2}$$
$$\therefore\,\,\,$$ $$\ell $$ = k G$${^{{1 \over 2}}}$$ $${\hbar ^{{1 \over 2}}}$$ $${C^{ - {3 \over 2}}}$$
= $${\left( {{{G\hbar } \over {{C^3}}}} \right)^{{1 \over 2}}}$$
(assume k = 1)
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