Question 195: Units , Dimensions and Error Analysis

Question

Which of the following combinations has the dimension of electrical resistance ($ \in $
0
is the permittivity of
vacuum and $\mu $
0
is the permeability of vacuum)?

Accepted Answer

According to Coulomb's law
F = ${1 \over {4\pi { \in _0}}}{{{q^2}} \over {{r^2}}}$
$ 	herefore $ ${ \in _0} = {1 \over {4\pi }}{{{q^2}} \over {F{r^2}}}$
$\left[ {{ \in _0}} ight] = {{{{\left[ {AT} ight]}^2}} \over {\left[ {ML{T^{ - 2}}} ight]\left[ {{L^2}} ight]}}$ = $\left[ {{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}} ight]$
Force between two parallel current carrying wires,
${F \over L} = {{{\mu _0}} \over {2\pi }}{{{i^2}} \over r}$
$ 	herefore $ ${{\mu _0}}$ = ${{2\pi rF} \over {{i^2}L}}$
$\left[ {{\mu _0}} ight] = {{\left[ L ight]\left[ {ML{T^{ - 2}}} ight]} \over {\left[ {{A^2}} ight]\left[ L ight]}}$ = $\left[ {ML{T^{ - 2}}{A^{ - 2}}} ight]$
From Ohm's law,
V = IR
$ 	herefore $ $R = {V \over I}$ $ = {U \over {It}} 	imes {1 \over I}$
[R] = ${{\left[ U ight]} \over {{{\left[ I ight]}^2}\left[ t ight]}}$ = ${{\left[ {M{L^2}{T^{ - 2}}} ight]} \over {\left[ {{A^2}} ight]\left[ T ight]}}$ = $\left[ {M{L^2}{T^{ - 3}}{A^{ - 2}}} ight]$
Let, R $ \propto $ ${\left[ {{ \in _0}} ight]^a}{\left[ {{\mu _0}} ight]^b}$
$ \Rightarrow $ $\left[ {M{L^2}{T^{ - 3}}{A^{ - 2}}} ight]$ =
$\left[ {{M^{ - 1}}{L^{ - 3}}{T^4}{A^2}} ight]$
$a$
$\left[ {ML{T^{ - 2}}{A^{ - 2}}} ight]$
b
$ \Rightarrow $ $\left[ {M{L^2}{T^{ - 3}}{A^{ - 2}}} ight]$ =
[ M
-$a$ + b
L
-3$a$ + b
T
4$a$ - 2b
A
2$a$ - 2b
]
By comparing both sides we get,
- $a$ + b = 1
- 3$a$ + b = 2
By solving we get,
$a$ = $ - {1 \over 2}$ and b = ${1 \over 2}$
$ 	herefore $ R = ${\left[ {{ \in _0}} ight]^{ - {1 \over 2}}}{\left[ {{\mu _0}} ight]^{{1 \over 2}}}$
= $\sqrt {{{{\mu _0}} \over {{ \in _0}}}} $

Answer

$\sqrt {{{{ \in _0}} \over {{\mu _0}}}} $

Answer

${{{{ \in _0}} \over {{\mu _0}}}}$

Answer

$\sqrt {{{{\mu _0}} \over {{ \in _0}}}} $

Answer

${{{{\mu _0}} \over {{ \in _0}}}}$

Units , Dimensions and Error Analysis

Which of the following combinations has the dimension of electrical resistance ( \in 0 is the permittivity of vacuum and \mu 0 is the permeability of vacuum)?

Choose the correct answer: