Question

Check whether the following relations are dimensionally correct :
(a) `V = (pi pr^(4))/(8 eta l)` (b) `h = (2 s)/(rho rg)`
(c ) `T = 2 pi sqrt((I)/(MB))` (d) `v = (E)/(B)`
Where `V:` volume, `p:` pressure, `r:` radius, `eta:` coefficient of viscosity, `l:` length, `h:` height, `s:` surface tension, `rho:` density, `g:` acceleration due to gravity, `I:` moment of inertia, `M:` magnetic moment, `B:` magnetic field, `v:` speed and `E:` electric field.

Answer 1

A relation will be dimensionally correct if dimensiosn on both sides are same.
(a) `LHS` dimesion : `[V] = L^(3)`
`RHS` dimension : `(pi pr^(4))/(8 eta l) = (ML^(-1) T^(-2) L^(4))/(ML^(-1) T^(-1) L) = L^(3) T^(-1)`
`LHS != RHS` not correct
(b) `RHS` dimension : `(2 s)/(rho rg) = (MT^(-2))/(ML^(-3) LLT^(-2)) = L`
`LHS` dimension : `h = L`
`LHS = RHS`, correct
(c ) `RHS` dimension :
`2 pi sqrt((I)/(MB)) = [ (ML^(2))/((AL^(2)) (MT^(-2) A^(-1)))]^(1)/(2) = T`
`LHS` dimension : `T`
`LHS = RHS`, correct
(d) `RHS` dimension : `(E)/(B) = (MLT^(-3))/(MT^(-2) A^(-1)) = LT^(-1)`
`LHS` dimension : `v = LT^(-1)`
`LHS = RHS`, correct