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Units check: What are the units of the permeability of free space μ₀ and the relative permeability μ_r?

Difficulty: Easy

Correct Answer: H/m for μ₀ and no units for μ_r

Explanation:


Introduction / Context:
Permeability links magnetic field intensity H to magnetic flux density B via B = μ H. Knowing the correct units prevents dimensional mistakes in electromagnetics and circuit analogies.



Given Data / Assumptions:

  • μ₀ is the absolute permeability of vacuum.
  • μ_r is the dimensionless factor describing a material’s permeability relative to vacuum: μ = μ₀ μ_r.
  • SI unit system (ampere, metre, kilogram, second).


Concept / Approach:
From B (tesla) = μ H (A/m), we obtain μ units as T·m/A. Since 1 T = 1 Wb/m² and 1 H = 1 Wb/A, T·m/A reduces to H/m. Therefore μ₀ has units henry per metre. Because μ_r is a ratio of two permeabilities, it is dimensionless (no units).



Step-by-Step Solution:

Start with B = μ H.Solve for μ: μ = B/H → (T) / (A/m) = T·m/A.Recognize T·m/A = (Wb/m²)·m/A = (Wb/A)/m = H/m.Thus μ₀ has units H/m and μ_r is unitless.


Verification / Alternative check:

Check with μ = μ₀ μ_r; multiplying H/m by a unitless μ_r leaves H/m.


Why Other Options Are Wrong:

Assigning H/m to μ_r gives dimensions to a pure ratio.Wb/m is not the correct derived unit for μ.


Common Pitfalls:

Confusing μ with reluctance or mistaking tesla–ampere conversions.


Final Answer:

H/m for μ₀ and no units for μ_r
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