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IF the charge of electron c , mass of el...

IF the charge of electron c , mass of electron m, speed of light in vaccum c and Planck's constant h are taken as fundamental quantities. Then the permeability of vaccum `mu_0` can be expressed as

A

`-h/(mc^2)`

B

`(hc)/(me^2)`

C

`h/(ce^2)`

D

`(mc^2)/(he^2)`

Text Solution

Verified by Experts

We can expressed the permeability of vaccum
`mu_0 prop e^a m^b c^c h^d`
or `mu_0=ke^a m^b c^c h^d`………….(i)
Where, K is a dimensional constant
As we know that
Dimension of `mu_0=[M L T^-2 A^(-2)]`,
`e=[A,T]`
`m=[M]`
`c=[L T^-1]`
and `h= [ M L^2 T^-1]`
Putting the dimension of various physical quantities in Eq (i) we get
`[M L T^-2 A^-2] =k [A T]^a[ M]^b [L T ^-1]^c ][ M L^2 T^-1]^d`
`[ M L T^-2 A ^-2]=k [M]^(b+d) [L]^(c+2d) [T]^(a-c-d) [A]^a`
Comparing the powers of M,L,T and A on the both sides, we get
`b+d=1`....(ii)
`c+2d=1`.....(iii)
`a-c-d=-2`.....(iv)
`a=-2` .....(v)
After solving the Eqs. (ii), (iii), (iv) (v) , we get
`a=-2`,
`b=0`,
`c=-1`
and `d=1`,
`thereforemu_0=h/(ce^2)`
So, the permeability of vaccum `mu_0` can be expressed as `h/(ce^2)`.
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    D
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