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Question: If \({{\text{ }\\!\\!\mu\\!\\!\text{ }}_{\text{o}}}\) is the permeability of free space and \({{\in ...

If  !!μ!! o{{\text{ }\\!\\!\mu\\!\\!\text{ }}_{\text{o}}} is the permeability of free space and ∈o{{\in }_{\text{o}}} is the permittivity of free space, the speed of light in vacuum is given by
a)μ∘∈∘ b)μ∘∈∘ c)1μ∘∈∘ d)∈∘μ∘ \begin{aligned} & a)\sqrt{{{\mu }_{\circ }}{{\in }_{\circ }}} \\\ & b)\sqrt{\dfrac{{{\mu }_{\circ }}}{{{\in }_{\circ }}}} \\\ & c)\sqrt{\dfrac{1}{{{\mu }_{\circ }}{{\in }_{\circ }}}} \\\ & d)\sqrt{\dfrac{{{\in }_{\circ }}}{{{\mu }_{\circ }}}} \\\ \end{aligned}

Explanation

Solution

Light is an electromagnetic wave which consists of alternating electric and magnetic fields both perpendicular to each other. The answer to the above question was proved by Maxwell. The value of permeability of free space i.e.  !!μ!! o=4π×10−7 Ns2C-2{{\text{ }\\!\\!\mu\\!\\!\text{ }}_{\text{o}}}=4\pi \times {{10}^{-7}}\text{ N}{{\text{s}}^{\text{2}}}{{\text{C}}^{\text{-2}}} and the value of permittivity of free space i.e. ∈o=8.85×10−12 C2N-1m-2{{\in }_{\text{o}}}=8.85\times {{10}^{-12}}\text{ }{{\text{C}}^{\text{2}}}{{\text{N}}^{\text{-1}}}{{\text{m}}^{\text{-2}}}. Hence we can substitute the values of μ∘ and ∈∘{{\mu }_{\circ }}\text{ and }{{\in }_{\circ }} in the options provided and see if the value equals to speed of light in vacuum.

Complete step by step answer:
So let us substitute the values of μ∘ and ∈∘{{\mu }_{\circ }}\text{ and }{{\in }_{\circ }} in the above options provided. The speed of light in vacuum is equal to 3×108ms−13\times {{10}^{8}}m{{s}^{-1}}
μ∘∈∘ 4π×10−7×8.85×10−12  33.34×10−10 \begin{aligned} & \sqrt{{{\mu }_{\circ }}{{\in }_{\circ }}} \\\ & \sqrt{4\pi \times {{10}^{-7}}\times 8.85\times {{10}^{-12}}\text{ }} \\\ & 33.34\times {{10}^{-10}} \\\ \end{aligned}
Hence option a is incorrect.
μ∘∈∘ 4π×10−7 8.85×10−12=3.76×10−3 \begin{aligned} & \sqrt{\dfrac{{{\mu }_{\circ }}}{{{\in }_{\circ }}}} \\\ & \sqrt{\dfrac{4\pi \times {{10}^{-7}}\text{ }}{8.85\times {{10}^{-12}}}}=3.76\times {{10}^{-3}} \\\ \end{aligned}
Hence option b is also incorrect.
1μ∘∈∘ 14π×8.85×10−19=3×108 \begin{aligned} & \sqrt{\dfrac{1}{{{\mu }_{\circ }}{{\in }_{\circ }}}} \\\ & \sqrt{\dfrac{1}{4\pi \times 8.85\times {{10}^{-19}}}}=3\times {{10}^{8}} \\\ \end{aligned}
The above value equals the speed of light in vacuum.
Hence the correct answer is option c.

Note:
The better and the easier way of solving the problem would have been just substituting the units of μ∘ and ∈∘{{\mu }_{\circ }}\text{ and }{{\in }_{\circ }} in the options provided. If the answer generated on substituting the units comes to meters per second then that would have been the correct answer. From the above result we can also conclude that the speed of light in vacuum is the highest, as in any other given medium the μ∘ and ∈∘{{\mu }_{\circ }}\text{ and }{{\in }_{\circ }} will be times the factor k where k( is not equal to 1) is the dielectric constant of the medium other than vacuum. From this result Maxwell was also able to conclude that the light is an electromagnetic wave. And every electromagnetic wave travels with constant velocity.