Question 180: 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: