The SI unit of permeability is:

Question

Prepp Team · Accepted Answer

Understanding the SI Unit of Permeability

Permeability is a fundamental property of a material that describes how easily it supports the formation of a magnetic field within itself when subjected to an external magnetic field. It is a measure of the degree of magnetization that a material obtains in response to an applied magnetic field.

The relationship between magnetic flux density ($\mathbf{B}$) and magnetic field strength ($\mathbf{H}$) in a material is given by:

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

where $\mu$ is the permeability of the material.

Deriving the SI Unit of Permeability

From the equation $\mathbf{B} = \mu \mathbf{H}$, we can express permeability as:

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

Let's consider the SI units of $\mathbf{B}$ and $\mathbf{H}$:

The SI unit of magnetic flux density ($\mathbf{B}$) is the Tesla (T).
The SI unit of magnetic field strength ($\mathbf{H}$) is Ampere per meter (A/m).

Substituting these units into the expression for $\mu$:

$$ \text{Unit of } \mu = \frac{\text{Unit of } \mathbf{B}}{\text{Unit of } \mathbf{H}} = \frac{\text{Tesla}}{\text{Ampere/meter}} = \frac{\text{Tesla} \cdot \text{meter}}{\text{Ampere}} $$

So, one possible SI unit for permeability is Tesla-meter per Ampere (T m/A).

Alternatively, we can derive the unit from the formula for the inductance of a component, like a solenoid. The inductance ($L$) of a long solenoid is given by:

$$ L = \frac{\mu N^2 A}{l} $$

where $\mu$ is the permeability of the core material, $N$ is the number of turns, $A$ is the cross-sectional area, and $l$ is the length of the solenoid.

Rearranging the formula to find $\mu$:

$$ \mu = \frac{Ll}{N^2 A} $$

Let's consider the SI units of the terms on the right side:

The SI unit of inductance ($L$) is the Henry (H).
The SI unit of length ($l$) is the meter (m).
The number of turns ($N$) is dimensionless.
The SI unit of area ($A$) is the square meter (m$^2$).

Substituting these units into the expression for $\mu$:

$$ \text{Unit of } \mu = \frac{(\text{Unit of } L) \cdot (\text{Unit of } l)}{(\text{Unit of } N)^2 \cdot (\text{Unit of } A)} = \frac{\text{Henry} \cdot \text{meter}}{(\text{dimensionless})^2 \cdot \text{meter}^2} = \frac{\text{Henry} \cdot \text{meter}}{\text{meter}^2} = \frac{\text{Henry}}{\text{meter}} $$

So, the SI unit for permeability is Henry per meter (H/m).

Equivalence of Units

Let's show that Tesla-meter per Ampere (T m/A) is equivalent to Henry per meter (H/m).

The Henry (H) is the unit of inductance. From Faraday's law ($V = -L \frac{dI}{dt}$ or $V \Delta t = -L \Delta I$), the unit of $L$ is $\frac{\text{Volt} \cdot \text{second}}{\text{Ampere}}$.
Magnetic flux ($\Phi$) has units of Volt-second (Wb). From Faraday's Law ($V = -\frac{d\Phi}{dt}$), $V \cdot s$ has units of flux. So, Henry (H) = $\frac{\text{Weber}}{\text{Ampere}}$.
Magnetic flux is also given by $\Phi = \int \mathbf{B} \cdot d\mathbf{A}$. So, the unit of Weber (Wb) is $\text{Tesla} \cdot \text{meter}^2$ (T m$^2$).
Therefore, Henry (H) = $\frac{\text{T m}^2}{\text{A}}$.
Now, let's look at Henry per meter (H/m):

$$ \frac{\text{Henry}}{\text{meter}} = \frac{\text{T m}^2 / \text{A}}{\text{meter}} = \frac{\text{T m}^2}{\text{A} \cdot \text{meter}} = \frac{\text{T m}}{\text{A}} $$

This confirms that Henry per meter (H/m) is indeed equivalent to Tesla-meter per Ampere (T m/A), which we derived from $\mathbf{B} = \mu \mathbf{H}$. Henry/meter is the standard SI unit used for permeability.

Analyzing the Options

Let's examine the given options based on our derivation:

Tesla: This is the SI unit of magnetic flux density ($\mathbf{B}$), not permeability ($\mu$). Incorrect.
Henry−meter: This means Henry multiplied by meter. The unit derived is Henry divided by meter. Incorrect.
Ampere turns: Ampere-turn is a unit of magnetomotive force (MMF), or related to magnetic field strength ($\mathbf{H}$) often given as Ampere-turns per meter. Incorrect.
Henry/meter: This is the SI unit of permeability ($\mu$), as derived from the inductance formula and shown to be equivalent to T m/A. Correct.

Therefore, the SI unit of permeability is Henry/meter.

Revision Table: Key Magnetic Units

Quantity	Symbol	SI Unit	Unit Symbol
Magnetic Flux Density	$\mathbf{B}$	Tesla	T
Magnetic Field Strength	$\mathbf{H}$	Ampere per meter	A/m
Permeability	$\mu$	Henry per meter	H/m
Magnetic Flux	$\Phi$	Weber	Wb
Inductance	$L$	Henry	H

Additional Information on Permeability

Permeability ($\mu$) is a crucial concept in understanding how magnetic fields behave in different materials. Here are some related points:

Vacuum Permeability ($\mu_0$): This is the permeability of free space (a vacuum). Its value is a fundamental constant, approximately $4\pi \times 10^{-7}$ H/m.
Relative Permeability ($\mu_r$): This is the ratio of the permeability of a material ($\mu$) to the permeability of free space ($\mu_0$), i.e., $\mu_r = \mu / \mu_0$. It is a dimensionless quantity.
Classification of Materials: Materials are classified based on their magnetic properties and relative permeability:
- Diamagnetic Materials: $\mu_r \lesssim 1$ (slightly less than 1). They weakly oppose the applied magnetic field.
- Paramagnetic Materials: $\mu_r \gtrsim 1$ (slightly greater than 1). They are weakly attracted to the applied magnetic field.
- Ferromagnetic Materials: $\mu_r \gg 1$ (much greater than 1). They are strongly attracted to the applied magnetic field and can become permanently magnetized.
Permeability plays a key role in determining the inductance of coils and the behavior of magnetic circuits.