Question

Do permeability and relative permeability have the same dimensions?

Answer 1

No they don't have same

The relative permeability μr is then defined as the ratio μ/μ0, which is dimensionless and has the same numerical value as the permeability in the cgs system

Answer 2

Explanation:

We know that the formula for magnetic permeability  

$\mu=\frac{B}{H}$

Where  

B = Magnetic Field

H = Magnetization Intensity

The SI unit of Magnetic Field is $N / A M$

Then its dimensional formula is  $\left[\begin{array}{lll}{M} & {A^{-1}} & {T^{-2}}\end{array}\right]$

But SI unit for Magnetization intensity is $A / M$

Its dimensional formula is  $\left[\begin{array}{ll}{A} & {L^{-1}}\end{array}\right]$

Thus the dimensional formula of permeability  

$\mu=\frac{\left[M \quad A^{-1} \quad T^{-2}\right]}{\left[A \quad L^{-1}\right]}$

Whereas relative permeability can be explained as the ratio of magnetic permeability of any medium to the permeability of the vacuum.

Where $\mu_0$ =absolute permeability of vacuum

So relative permeability is a dimensionless quantity

Therefore, the relative permeability is dimensionless quantity a permeability have different dimensions