Question 199: 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 c3

Answer

G2 $$\hbar $$ 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: