Units check: What are the units of the permeability of free space μ₀ and the relative permeability μ_r?
Electronics and Communication Engineering
Materials and Components
Difficulty: Easy
Choose an option
-
AH/m for both
-
BH/m for μ_r and no units for μ₀
-
CH/m for μ₀ and no units for μ_r
-
DWb/m for μ₀ and no units for μ_r
-
ET·m/A for μ_r and A·m for μ₀
Answer
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