Question

reciprocal of dimensions of square root of product of permitivity of free space and permiabilty of free space is
a)velocity
b)time
c)capacitance

Answer 1

With the magnetic permeability established, the electric permittivity takes the value given by the relationship 

c = 1/sqrt(ε0μ0) 

where the speed of light is given by 

c=2.99792458 x 10^8 m/s (exact) 

This gives a value of free space permittivity 

ε0 = 8.854187817 x 10^-12 F/m 

which in practice is often used in the form 

k = 1/4piε0 = 8.987552 X10^9 Nm^2/C^2 = Coulomb's constant 

So, product of permeability of free space and permittivity is 1/c^2.

Answer 2

The unknown quantity is (a) velocity.

Explanation:

The electrostatic force between two charges $Q$ and $q$ separated by a distance $R$ is given by

$F=\frac{1}{4\pi E_{o} } \frac{Qq}{R^{2} }$

Where, ε$_{o}$ is the permittivity of free space.

Hence, permittivity ε$_{o}$ will be

ε$_{o}$$=\frac{Qq}{F4(\pi )R^{2} }$  

We know, the SI unit of

$Q=As$     $;$ $F=kgm/s^{2}$    $;$  $R=m$

Hence, we get the dimension of ε$_{o}$ as

ε$_{o}$ $=\frac{A^{2} s^{2} }{(kgm/s^{2} )(m^{2} )}$

ε$_{o}$ $=A^{2} .s^{4} .m^{-3} .kg^{-1}$

Step 2

Now,

The Biot Savart's law for a long thin conducting wire is given by

$B=\frac{u_{o} }{4\pi } \frac{I.dlsin\alpha }{R^{2} }$

Hence, the permeability of free space μ$_{o}$ will be,

μ$_{o}$ $=\frac{B.R^{2} (4\pi )}{I.dl.(sin\alpha )}$

We know, the dimensions of

$B=kg.s^{-2} .A^{-1}$   $;$ $R=m$  $;$ $l=m$   $;$ $I=A$

Substituting the values, we get the dimensions for the permeability as

μ$_{o}$ $=\frac{kg.s^{-2} A^{-1} (m^{2} )}{A.(m)}$

μ$_{o}$ $= kg.s^{-2} .A^{-2} .m$

Step 3

Now, According to the question,

The square root of reciprocal of both the constants lets say $k$.

$k=\sqrt{\frac{1}{u_{o} E_{o} }$

Substituting the values of dimensions, we get

$k=\sqrt{\frac{1}{(kg.s^{-2} .A^{-2} .m)(A^{2} .s^{4} .m^{-3} .kg^{-1} )} }$

$k=\frac{1}{\sqrt{(s^{4-2}. m^{-3+1} )} }$

$k=\sqrt{\frac{m^{2} }{s^{2} }$

$k=m/s$

Hence, the dimension of the quantity which equals to square root of reciprocal of the permeability and permittivity of free space is m/s.

m/s is the dimension of only one quantity that is velocity.

Final answer:

Hence, the unknown quantity is (a) velocity.