In terms of potential difference `C`, electric current`I` , permittivity `epsilon_(0)`, permeability `mu_(0)` and speed of light `c`, the dimensionally correct equation `(s)` is `(are)`

Options

A. `mu_0 I^2 = epsilon_0 V^2`
B. `epsilon_0 I = mu_0 V`
C. `I = epsilon_0 cV`
D. `mu_0 cI = epsilon_0V`

Answer

(a,c)

Explanation

Here, Dimnsional formula for
`(i) V =ML^2 T^(-3) A^(-1)`
`(ii) I= [A^1]`
(iii) `c= [LT^(-1)]`
(iv)`epsilon_0 = [M^(-1) L^(-3) T^4 A^2]`
(v) `mu_0 =[M^1 L^1 T^(-2) A^(-2)]`
(a) `mu_o I^2 =epsilon_0 V^2`
`MLT^(-2) A^(-2)xxA^2 = M^(-1) L^(-3) T^4 A^2xx(ML^2 T^(-3) A^(-1))^2`
`MLT^(-2) = MLT^(-2)`
Hence option (a) is correct.
(b) `epsilon_0I =mu_0V`
`M^(-1)L^(-3) T^2 A^2xxA = M^1 L^1 T^(-2) A^(-2)xxML^2 T^(-3)A^(-1)`
`M^2 L^3 T^(-5) A^(-3)`
`:. LHS =! RHS, dimensionally.`
Hence option (b) is incorrect.
(c ) `I = epsilon_0 cV`
`A = M^(-1) L^(-3) T^4 A^2 xx LT^(-1)xxML^2 T^(-3) A^(-1) = A`
L.H.S. = R.H.S., dimensionally.
Hence option (c ) is correct.
(d) `mu_0 cI = epsilon_0 V`
`M^1 L^1 T^(-2) A^(-2) xx LT^(-1) xxA`
`=M^(-1) L^(-3) T^4A^2 xx ML^2 T^(-3) A^(-1)`
`M^1 L^2 T^(-3)A^(-1) =! M^0 L^(-1) T^1 A^1`
L.H.S =! R.H.S.
Hence option (d) is incorrect.