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_0I^2=in_0V^2`
B. `in_0I=mu_0V`
C. `I=in_0cV`
D. `mu_0cI=in_0V`

Answer

(a,c)

Explanation

Dimensions of `in_0=[M^-1L^-3T^4I^2]`,
`mu_0=[MLT^-2I^-2] ,V=[ML^2T^-3I^-1]`
(a) L.H.S, `mu_0I^2=(MLT^-2I^-2)=[MLT^-2]`
R.H.S, `in_0V^2=(M^-1L^-3T^4I^2)`
`x(M^2L^4T^-6I^-2)=[MLT^-2]`
`[L.H.S]=[R.H.S]`, hence option (a) is true.
(b) L.H.S., `in_0I=(M^-1L^-3T^4I^2)I`
`=[M^-1L^3T^4I^3]`
R.H.S, `mu_0V= [MLT^-2I^-2]xx[ML^2T^-3I^-1]`
`=[M^2L^3T^-5I^-3]`
`[L.H.S]!=[R.H.S]`, hence option (b) is false.
(c) L.H.S., `|I|=I`
R.H.S., `in_0CV=[M^-1L^-3T^4I^2][LT^-1]`
`x[ML^2T^-3I^-2]=I`
`[L.H.S]=[R.H.S]`, so, option (c) is true.
(d) L.H.S,. `mu_0CI=[MLT^-2I^-2][LT^-1][I]`
`[ML^2T^-3I^-1]`
R.H.S., `in_0V=[M^-1L^-3T^4I^2]`
`[ML^2T^-3I^-1]=[M^0L^-1TI]`
[L.H.S]!=[R.H.S], Thus, option (d) is wrong.