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Question: The value of \(\dfrac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }}\) is A) \(3 \times {10^6}{\text{m}}...

The value of 1μ0ε0\dfrac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }} is
A) 3×106ms−13 \times {10^6}{\text{m}}{{\text{s}}^{ - 1}}
B) 3×108ms−13 \times {10^8}{\text{m}}{{\text{s}}^{ - 1}}
C) 3×104ms−13 \times {10^4}{\text{m}}{{\text{s}}^{ - 1}}
D) 322ms−1322{\text{m}}{{\text{s}}^{ - 1}}

Explanation

Solution

Electromagnetic waves are considered to be synchronous oscillations of electric field oscillations and magnetic field oscillations. Both the electric field oscillations and the magnetic field oscillations are perpendicular to each other and are perpendicular to the direction of propagation of the electromagnetic wave. In the given quantity, μ0{\mu _0} denotes the permeability of free space and is related to the magnetic field component while ε0{\varepsilon _0} denotes the permittivity of free space and is related to the electric field component.

Complete step by step answer.
Step 1: List the values of μ0{\mu _0} and ε0{\varepsilon _0} to obtain the value of the given term.
The permittivity of a material refers to the capability of that material to allow or transmit electric fields. The permittivity of free space is represented by ε0{\varepsilon _0} and its value is known to be ε0=8⋅85×10−12C2N−1m2{\varepsilon _0} = 8 \cdot 85 \times {10^{ - 12}}{{\text{C}}^2}{{\text{N}}^{ - 1}}{{\text{m}}^2}.
Similarly, the permeability of a material refers to the ability of that material to allow the entry of magnetic field lines in the material. It generally describes the ability of magnetization possessed by the material for the applied magnetic field. The permeability of free space is represented by μ0{\mu _0} and its value is known to be μ0=4π×10−7NA−2=4π×10−7NC−2s2{\mu _0} = 4\pi \times {10^{ - 7}}{\text{N}}{{\text{A}}^{ - 2}} = 4\pi \times {10^{ - 7}}{\text{N}}{{\text{C}}^{ - 2}}{{\text{s}}^2}.
We have to determine the value of 1μ0ε0\dfrac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }}. So we can substitute the values μ0=4π×10−7NA−2{\mu _0} = 4\pi \times {10^{ - 7}}{\text{N}}{{\text{A}}^{ - 2}} and ε0=8⋅85×10−12C2N−1m2{\varepsilon _0} = 8 \cdot 85 \times {10^{ - 12}}{{\text{C}}^2}{{\text{N}}^{ - 1}}{{\text{m}}^2} to obtain the value of the given quantity.
i.e., 1μ0ε0=1(4π×10−7NC−2s2)×(8⋅85×10−12C2N−1m2)=3×108ms−1\dfrac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }} = \dfrac{1}{{\sqrt {\left( {4\pi \times {{10}^{ - 7}}{\text{N}}{{\text{C}}^{ - 2}}{{\text{s}}^2}} \right) \times \left( {8 \cdot 85 \times {{10}^{ - 12}}{{\text{C}}^2}{{\text{N}}^{ - 1}}{{\text{m}}^2}} \right)} }} = 3 \times {10^8}{\text{m}}{{\text{s}}^{ - 1}}
∴\therefore the value of 1μ0ε0\dfrac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }} is obtained to be 3×108ms−13 \times {10^8}{\text{m}}{{\text{s}}^{ - 1}}.

So the correct option is B.

Note: Alternate method
The given quantity actually represents the speed of the electromagnetic wave in free space i.e., 1μ0ε0=c\dfrac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }} = c.
We know that the speed of electromagnetic wave which is basically light has a speed c=3×108ms−1c = 3 \times {10^8}{\text{m}}{{\text{s}}^{ - 1}} in free space. So the value of the given quantity 1μ0ε0\dfrac{1}{{\sqrt {{\mu _0}{\varepsilon _0}} }} will be 3×108ms−13 \times {10^8}{\text{m}}{{\text{s}}^{ - 1}} and the correct option will be B.