(a) In the formula `X=3YZ^(2)`, `X` and `Z` have dimensions of capcitnce and magnetic inlduction, respectively. What are the dimensions of `Y` in `MKSQ` system?
(b) `A` qunatity `X` is given by `epsilon_(0)` `L` `((Delta)V)/((Delta)r)`, where `epsilon_(0)` is the permittivity of free space, `L` is a lenght, `DeltaV` is a potential difference and `Deltat` is a time interval. Find the dimensions of `X`.
(c) If `E,M,J` and `G` denote energy , mass , angular momentum and gravitational constant, respectively. find dimensons of `(E J^(2))/(M^(5) G^(2))`
(d) If `e,h,c` and `epsilon_(0)` are electronic charge, Planck 's constant speed of light and permittivity of free space. Find the dimensions of `(e^(2))/(2epsilon_(0)hc)`.

Explanation

(a) `X=3XYZ^(2)rArrY=(X)/(3Z^(2))`
`[Y]=(M^(-1)L^(-2)T^(4)A^(2))/([MT^(-2) A^(-1)]^(2))=M^(-3)L^(-2)T^(8)A^(4)`
charge Q=It
`[Q]=AT` ,
`=M^(-3) L^(-2) T^(4)(AT)6(4)`
`=M^(-3)L^(-2)T^(4)Q^(4))`
(b) `X=epsilon_(0) L (Deltav)/(Deltat)`
`[X]=([M^(-1)L^(-3)T^(4)A6(2)][L][ML^(2)T^(-3)A^(-1)])/([T])=A`
(c) `(EJ^(2))/(M^(5) G^(2))=([ML^(2)T^(-2)][ML^(2)T^(-1)]^(2))/([M]^(5)[M^(-1)L^(3)T^(-2)]^(2))=([M^(3)L^(6)T^(-4)])/([M^(3)L^(6)T6(-4)])`
(d) `(e^(2))/(2epsilon_(0)hc)=([AT^(2)])/([M^(-1)L^(-3)T^(4)A^(2)][ML^(2)T^(-1)][LT^(-1)])`