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  • If electronic charge e, electron mass m, speed of light in vacuum c and Planck’s constant h are taken as fundamental quantities, the permeability of vacuum <img alt="\mu _{0}" src="http://ent

If electronic charge e, electron mass m, speed of light in vacuum c and Planck’s constant h are taken as fundamental quantities, the permeability of vacuum \mu _{0} can be expressed in units of :

Option: 1

\left ( \frac{hc}{me^{2}} \right )


Option: 2

\left ( \frac{h}{me^{2}} \right )


Option: 3

\left ( \frac{h}{ce^{2}} \right )


Option: 4

\left ( \frac{mc^{2}}{he^{2}} \right )


Answers (1)

best_answer

\\ \text{The dimensions of permeability of vacuum in terms of other fundamental quantities can be written as,}\\ \left[\mu_{0}\right]=\left[e^{a} c^{b} h^{c} m^{d}\right] \ldots(1) \\ \text{Dimensions of} \ \mu_{0} \ are \left[M L T^{-2} A^{-2}\right]. \\ \text{Dimensions of} \ e \ are [AT]. \\ \text{Dimensions of} \ \mathrm{c} \ are \left[L T^{-1}\right]. \\ \text{Dimensions of} \ h \ are \left[M L^{2} T^{-1}\right]. \\ \text{Dimensions of} \ \mathrm{m} \ are [M]. \\ \text{Substituting all the values in the equation (1) we get} , \\ \left[M L T^{-2} A^{-2}\right]=[A T]^{a}\left[L T^{-1}\right]^{b}\left[M L^{2} T^{-1}\right]^{c}[M]^{d} \\ \left[M L T^{-2} A^{-2}\right]=\left[A^{a} L^{b+2 c} T^{a-b-c} M^{c+d}\right]

\\ \text{Equating the powers on both the sides we get,} \\ c+d=1 \ldots(2) \\ b+2 c=1 \ldots(3) \\ a-b-c=-2 \ldots (4) \\ a=-2 \ldots(5) \\ \text{Substituting value of a in equation. (4) we get,} \\ b+c=0 \ldots(6) \\ \text{Subtracting equation. (6) and (3) we get,} \\ c=1 \\ \text{Substituting value of a and c in equation. (4) we get,} \\ b=-1

\\ \text{Now, substituting the value of}\ \mathrm{c} \ \text{in the equation. (2) we get,} \\ d=0 \\ \text{Substituting these values in the equation. (1) we get, }\\ \left[\mu_{0}\right]=\left[e^{-2} c^{-1} h^{1} m^{0}\right] \\ \\ \Rightarrow\left[\mu_{0}\right]=\left[e^{-2} c^{-1} h^{1}\right] \\ \Rightarrow \mu_{0}=\frac{h}{c e^{2}} \\ \text{Hence, the permeability of vacuum} \ \mu_{0} \ \text{can be expressed in units of} \left(\frac{h}{c e^{2}}\right)

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shivangi.bhatnagar

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