Magnetic flux density

Prof. Dr. Thorsten Jungmann (last updated 2025-09-04)

The magnetic flux density (German: magnetische Flussdichte) describes the strength of the magnetic field at a particular point in space. It characterizes the density of the magnetic field lines and is a measure of how strongly the magnetic field acts on moving electric charges and current-carrying conductors.

Vector quantity in a homogeneous field → scalar quantity

This is a vector quantity. In this script, however, it is treated by convention as a scalar quantity in the context of the homogeneous field (for a detailed justification).

1 Quantity symbol and unit

The quantity symbol of the magnetic flux density is $\vec{B}$, or $B$ as a scalar quantity.

Its dimension (unit) is the tesla, named after Nikola Tesla, abbreviated with the unit symbol $\mathrm{T}$.

$$[B]=\mathrm{T}=\frac{\mathrm{Wb}}{{\mathrm{m}}^2}=\frac{\mathrm{V}\cdot \mathrm{s}}{{\mathrm{m}}^2}$$

To denote a position-dependent magnetic flux density, the quantity symbol $\vec{B}(\vec{r})$ is used for the flux density vector, since the magnetic flux density is a vector quantity. Without the vector property, the magnitude is meant.

2 Practical and worked examples

2.1 Relationship with magnetic flux

For a homogeneous magnetic field perpendicular to the area, the following applies:

$$B=\frac{\Phi }{A}$$

where:

• $\Phi$ the magnetic flux
• $A$ the effective area

Example: $\Phi =2.5\mathrm{mWb}$, $A=5.0\cdot 10^{-4}{\mathrm{m}}^2$

$$B=\frac{2.5\cdot 10^{-3}\mathrm{Wb}}{5.0\cdot 10^{-4}{\mathrm{m}}^2}=\underline{5.0\mathrm{T}}$$

2.2 Relationship with magnetic field strength

$$\vec{B}=\mu \cdot \vec{H}={\mu }_0\cdot {\mu }_r\cdot \vec{H}$$

where:

• $\mu$ the permeability
• ${\mu }_0$ the vacuum permeability (${\mu }_0=4\pi \cdot 10^{-7}\frac{\mathrm{V}\cdot \mathrm{s}}{\mathrm{A}\cdot \mathrm{m}}$)
• ${\mu }_r$ the relative permeability of the material
• $H$ the magnetic field strength

Example: $H=1000\frac{\mathrm{A}}{\mathrm{m}}$, ${\mu }_r=2000$ (iron)

$$B=4\pi \cdot 10^{-7}\cdot 2000\frac{\mathrm{V}\cdot \mathrm{s}}{\mathrm{A}\cdot \mathrm{m}}\cdot 1000\frac{\mathrm{A}}{\mathrm{m}}\approx 2.51\frac{\mathrm{V}\cdot \mathrm{s}}{{\mathrm{m}}^2}=\underline{2.51\mathrm{T}}$$

2.3 Force on a current-carrying conductor

The force on a straight, current-carrying conductor in a homogeneous magnetic field is:

$$\vec{F}=I\cdot \vec{l}\times \vec{B}$$

For a homogeneous magnetic field perpendicular to the conductor:

$$F=B\cdot I\cdot l$$

Example: $B=0.5\mathrm{T}$, $I=2.0\mathrm{A}$, $l=0.10\mathrm{m}$

$$F=0.5\mathrm{T}\cdot 2.0\mathrm{A}\cdot 0.10\mathrm{m}=0.10\frac{\mathrm{V}\cdot \mathrm{s}\cdot \mathrm{A}\cdot \mathrm{m}}{{\mathrm{m}}^2}=0.10\mathrm{N}$$

Unit conversion to $\mathrm{N}$:

$$\frac{\mathrm{V}\cdot \mathrm{s}\cdot \mathrm{A}}{\mathrm{m}}=\frac{\mathrm{J}\cdot \mathrm{A}}{\mathrm{C}\cdot \mathrm{m}}=\frac{\mathrm{N}\cdot \mathrm{m}\cdot \mathrm{A}}{\mathrm{A}\cdot \mathrm{s}\cdot \mathrm{m}}=\mathrm{N}$$

2.4 Induced voltage

According to the law of induction, a change in the flux density over time gives:

$$u(t)=-N\cdot A\cdot \frac{\mathrm{d}B}{\mathrm{d}t}$$

Example: $N=100$, $A=1.0\cdot 10^{-3}{\mathrm{m}}^2$, $\frac{\mathrm{d}B}{\mathrm{d}t}=50T/s$

$$u=-100\cdot 1.0\cdot 10^{-3}{\mathrm{m}}^2\cdot 50\frac{\mathrm{T}}{\mathrm{s}}$$ $$u=-5.0{\mathrm{m}}^2\cdot \frac{\mathrm{T}}{\mathrm{s}}$$

With $\mathrm{T}=\frac{\mathrm{V}\cdot \mathrm{s}}{{\mathrm{m}}^2}$:

$$u=-5.0{\mathrm{m}}^2\cdot \frac{\mathrm{V}\cdot \mathrm{s}}{{\mathrm{m}}^2\cdot \mathrm{s}}=-5.0\mathrm{V}$$

3 Typical orders of magnitude

While the Earth’s magnetic field has a flux density of about $25$ to $65\mu \mathrm{T}$, permanent magnets reach values of $0.1$ to $1.5\mathrm{T}$. Electromagnets can reach $2$ to $3\mathrm{T}$, and superconducting magnets even more than $10\mathrm{T}$. In transformers, $1.0$ to $1.8\mathrm{T}$ is typically used.

4 Example phrasings

• ‘The magnetic flux density in the air gap is $0.8\mathrm{T}$.’
• ‘The permanent magnet produces a flux density of $1.2\mathrm{T}$.’
• ‘At a flux density of $2.0\mathrm{T}$, the saturation of the iron core begins.’
• ‘The Earth’s magnetic field has a flux density of about $50\mu \mathrm{T}$.’

5 Related quantities

• The magnetic flux is the surface integral of the flux density:

$$\Phi ={\int }_A\vec{B}\cdot \mathrm{d}\vec{A}$$

• The magnetic flux density is linked to the magnetic field strength via the permeability:

$$\vec{B}={\mu }_0\cdot {\mu }_r\cdot \vec{H}$$

• The force on a current-carrying conductor:

$$\vec{F}=I\cdot \vec{l}\times \vec{B}$$

• The induced voltage for a change over time:

$$u(t)=-N\cdot \frac{\mathrm{d}\Phi }{\mathrm{d}t}=-N\cdot A\cdot \frac{\mathrm{d}B}{\mathrm{d}t}$$

6 Related components

• Inductor: generation and concentration of the magnetic flux density
• Transformer: transfer by magnetic flux density
• Electric motor: force effect by magnetic flux density
• Relay and contactor: switching operation by magnetic force