Question

The relation between \[\left[ {{ \in _0}} \right]\]and\[\left[ {{\mu _0}} \right]\] is
A. \[\left[ {{\mu _0}} \right] = \left[ {{ \in _0}} \right]{\left[ L \right]^2}{\left[ T \right]^{ - 2}}\]
B. \[\left[ {{\mu _0}} \right] = \left[ {{ \in _0}} \right]{\left[ L \right]^{ - 2}}{\left[ T \right]^2}\]
C. \[\left[ {{\mu _0}} \right] = {\left[ {{ \in _0}} \right]^{ - 1}}{\left[ L \right]^2}{\left[ T \right]^{ - 2}}\]
D. \[\left[ {{\mu _0}} \right] = {\left[ {{ \in _0}} \right]^{ - 1}}{\left[ L \right]^{ - 2}}{\left[ T \right]^2}\]

Answer 1

Hint: Use the Coulomb’s law and Biot-Savart’s law to find the relation between the given constants.

Complete step by step solution:
From Coulomb’s law we have \[\dfrac{1}{{4\pi { \in _0}}} = 9 \times {10^9}\]
\[ \Rightarrow 4\pi { \in _0} = \dfrac{1}{{9 \times {{10}^9}}} \Rightarrow { \in _0} = \dfrac{1}{{9 \times {{10}^9} \times 4\pi }}\]
Where, \[{ \in _0}\] is the absolute electrical permittivity of the free space.

From Biot-Savart’s law, we have
\[\dfrac{{{\mu _0}}}{{4\pi }} = {10^{ - 7}} \Rightarrow {\mu _0} = {10^{ - 7}} \times 4\pi \]
Where,\[{\mu _0}\]is the absolute permeability of free space. Its value depends on the system of units chosen for the measurement of various quantities and also on the medium between a point and the electric current.

Multiplying above two equations we get, \[{\mu _0} \times { \in _0} = {10^{ - 7}} \times 4\pi \times \dfrac{1}{{9 \times {{10}^9} \times 4\pi }}\]
\[ \Rightarrow {\mu _0} \times { \in _0} = \dfrac{1}{{9 \times {{10}^{16}}}} \Rightarrow {\mu _0} \times { \in _0} = \dfrac{1}{{{{\left( {3 \times {{10}^8}} \right)}^2}}}\]
\[ \Rightarrow {\mu _0} \times { \in _0} = \dfrac{1}{{{{\left( c \right)}^2}}}\]
Where, c is the velocity of light. The unit for the speed of light is \[m{s^{ - 2}}\]. So, the dimensions of velocity of light is\[L{T^{ - 2}}\]\[ \Rightarrow {\mu _0} \times { \in _0} = \dfrac{1}{{L{T^ - }^2}} \Rightarrow {\mu _0} = {\left[ {{ \in _0}} \right]^{ - 1}}{\left[ L \right]^ - }^2{\left[ T \right]^2}\]

Hence, the correct option is (D).

Note: According to Coulomb’s law the force of interaction between any two point charges is directly proportional to the product of the charges and inversely proportional to the distance between them. Biot-Savart’s law is an experimental law predicted by Biot and Savart . This law deals with the magnetic field induction at a point due to a small current element.The product of absolute permittivity and absolute permeability of space is related to the velocity of light. This relation forms the basis of the theory of relativity. Einstein used this equation to find the velocity of light. It is the maximum speed in space by which a body can travel. No body in space can travel with a velocity greater than the velocity of light.