Magnetic field strength

Prof. Dr. Thorsten Jungmann (last updated 2026-05-04)

The magnetic field strength (German: magnetische Feldstärke) describes the ‘driving’ excitation of the magnetic field by currents. It is generated by an inductor. Via the permeability, it is linked to the magnetic flux density.

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 field strength is the capital letter $H$.
Its unit is ampere per metre with the unit symbol $A/m$.

$$[H]=A/m$$

2 Practical and worked examples

Basic relationships:

$$\vec{B}=\mu \vec{H},\mu ={\mu }_0{\mu }_r$$ $$\Theta =\oint \vec{H}\cdot \mathrm{d}\vec{l}=N\cdot I$$

2.1 Long solenoid (interior, fringe fields neglected)

$$H\approx \frac{NI}{l}$$

Example: $N=500$, $I=0.20,\mathrm{A}$, $l=0.25,\mathrm{m}$

$$H\approx \frac{500\cdot 0.20}{0.25}\frac{\mathrm{A}}{\mathrm{m}}=400A/m$$

For ${\mu }_r=2000$ with ${\mu }_0=4\pi \cdot 10^{-7}\frac{\mathrm{N}}{{\mathrm{A}}^2}$:

$$B={\mu }_0{\mu }_{r}H\approx 1.0\mathrm{T}$$

2.2 Air gap in the magnetic circuit (gap dominates)

In the air gap, the following applies approximately:

$$H_{\mathrm{gap}}\approx \frac{B}{{\mu }_0},\Theta \approx H_{\mathrm{gap}}l$$

Example: desired flux density $B=0.50\mathrm{T}$, gap $l=1.0\mathrm{mm}$

$$H_{\mathrm{gap}}\approx \frac{0.50\mathrm{T}}{1.256\cdot 10^{-6}\frac{\mathrm{N}}{{\mathrm{A}}^2}}\approx 3.98\cdot 10^{5}A/m$$

Unit check:

$$[H]=\frac{[B]}{[\mu ]}=\frac{T}{\frac{N}{A^2}}=\frac{\frac{kg}{A\cdot s^2}}{\frac{N}{A^2}}=\frac{kg\cdot A^2}{A\cdot s^2\cdot N}=\frac{A}{m}$$

Magnetomotive force:

$$\Theta \approx 3.98\cdot 10^5\cdot 1.0\cdot 10^{-3}\approx 398\mathrm{A}$$

With $N=200$ turns, the current $I$ required to achieve the desired magnetic flux density of $0.5\mathrm{T}$ is:

$$I=\frac{\Theta }{N}\approx 2.0\mathrm{A}$$

2.3 From flux and cross-section

For a homogeneous cross-section:

$$B=\frac{\Phi }{A},H=\frac{B}{\mu }$$

Example: $\Phi =20\mu \mathrm{Wb}$, $A=1.0\cdot 10^{-4}{\mathrm{m}}^2$, air $(\mu ={\mu }_0)$

$$B=0.20\mathrm{T},H\approx \frac{0.20}{1.256\cdot 10^{-6}}\frac{T}{\frac{N}{A^2}}\approx 1.59\cdot 10^{5}A/m$$

3 Typical orders of magnitude

• Earth’s magnetic field ($B\approx 25\dots 65\mu \mathrm{T}$): $H\approx 20\dots 50A/m$
• Small electromagnets/relays (with air gap): $H\approx 10^3\dots 10^{5}A/m$
• Strong fields in the air gap (e.g. $B=1.5\mathrm{T}$): $H\approx 1.2\cdot 10^{6}A/m$

4 Example phrasings

• ‘In the solenoid, $H=NI/l$ results.’
• ‘In the air gap, $H=B/{\mu }_0$ applies approximately.’
• ‘The mean field strength along the closed path is $H\approx \Theta /l$.’

5 Related quantities and relationships

• Magnetic flux density:
  $\vec{B}=\mu \vec{H}$
• Magnetomotive force:
  $\Theta =\oint \vec{H}\cdot \mathrm{d}\vec{l}=N\cdot I$
• Magnetic flux and reluctance:
  $\Phi ={\int }_A\vec{B}\cdot \mathrm{d}\vec{A},\Phi =\frac{\Theta }{R_{\mathrm{m}}},R_{\mathrm{m}}=\frac{l}{\mu A}$
• Flux linkage and inductance:
  $\Lambda =N\Phi ,L=\frac{\Lambda }{I}$

6 Related components

• Inductor
• Relay and contactor
• Transformer