Question 214: Units , Dimensions and Error Analysis

Question

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 ?

Accepted Answer

Plank length,
$\ell $ = k G
p
$\hbar $
q
C
r
[ M
o
L T
o
] = [ M
$-$1
L
3
T
$-$2
]
p
[ M L
2
T
$-$1
]
q
[ L T
$-$1
]
r
[M
o
L T
o
] = [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 = ${1 \over 2},$ $r =  - {3 \over 2}$
$	herefore\,\,\,$ $\ell $ = k G${^{{1 \over 2}}}$ ${\hbar ^{{1 \over 2}}}$ ${C^{ - {3 \over 2}}}$
= ${\left( {{{G\hbar } \over {{C^3}}}} ight)^{{1 \over 2}}}$
(assume k = 1)

Answer

G $\hbar $
2
c
3

Answer

G
2
$\hbar $ c

Answer

${G^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}}}{\hbar ^2}c$

Answer

${\left( {{{G\hbar } \over {{c^3}}}} \right)^{{\raise0.5ex\hbox{$\scriptstyle 1$}
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{$\scriptstyle 2$}}}}$

Units , Dimensions and Error Analysis

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 ?

Choose the correct answer: