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Question: The dimensions of \[{{\left( {{\mu }_{0}}{{\varepsilon }_{0}} \right)}^{{-1}/{2}\;}}\]are: (A) \[\...

The dimensions of (μ0ε0)1/2  {{\left( {{\mu }_{0}}{{\varepsilon }_{0}} \right)}^{{-1}/{2}\;}}are:
(A) [L1T]\left[ {{L}^{-1}}T \right]
(B) [LT1]\left[ L{{T}^{-1}} \right]
(C) [L1/2  T1/2  ]\left[ {{L}^{{1}/{2}\;}}{{T}^{{1}/{2}\;}} \right]
(D) [LT1/2  ]\left[ L{{T}^{{1}/{2}\;}} \right]

Explanation

Solution

In the given equation, quantities μ0{{\mu }_{0}}and ε0{{\varepsilon }_{0}}are magnetic constant or the permeability of free space and electric constant or permittivity of free space. The direct dimensions of these quantities are not known but they were obtained by Maxwell from the electromagnetic laws. It is related to the velocity of the light in vacuum.

Complete answer:
The oscillation of electric field and magnetic field in vacuum creates electromagnetic waves. The light is electromagnetic waves in free space. The velocity of the light is denoted by cc.The velocity of light cc is constant therefore cc depends on electric and magnetic constant. The electric constant is known by the permeability of free space (μ0{{\mu }_{0}}) and magnetic constant is known by permittivity of free space (ε0{{\varepsilon }_{0}}).

c=1(μ0ε0)c=\dfrac{1}{\sqrt{\left( {{\mu }_{0}}{{\varepsilon }_{0}} \right)}}
(μ0ε0)1/2  =c\therefore {{({{\mu }_{0}}{{\varepsilon }_{0}})}^{{-1}/{2}\;}}=c

The unit of the cc is m/s  {m}/{s}\;. So the unit of given quantity is m/s  {m}/{s}\;.
The dimensional equivalent of unit m/s  {m}/{s}\; is [LT1]\left[ L{{T}^{-1}} \right]

So, the correct answer is “Option B”.

Additional Information:
Some important values if these constants:
Permeability of free space (μ0{{\mu }_{0}}) = 4π×107NA24\pi \times {{10}^{-7}}N{{A}^{-2}}
Permittivity of free space (ε0{{\varepsilon }_{0}}) = 8.85419×1012C2N1m28.85419\times {{10}^{-12}}{{C}^{2}}{{N}^{-1}}{{m}^{-2}}
Velocity of light (cc) = 2.99792×108ms12.99792\times {{10}^{8}}m{{s}^{-1}}

Note:
The dimensional relationship does not mean that the relation between both side quantities are linearly the same, it only compares the dimensions. Here the relation between the speed of light and (μ0ε0)({{\mu }_{0}}{{\varepsilon }_{0}}) is not only inversely proportional but the square root of (μ0ε0)({{\mu }_{0}}{{\varepsilon }_{0}}) inversely proportional to speed of light but the dimensional does not predict it.