Magnetic flux

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

Magnetic flux (German: magnetischer Fluss) describes how much magnetic field passes through an area. Mathematically, it is the surface integral of the magnetic flux density over an area with a given orientation.

Analogy: magnetic circuit/electric circuit

Magnetic flux corresponds to the electric current in an electric circuit.

1 Quantity symbol and unit

The quantity symbol of magnetic flux is the Greek capital letter $\Phi$ (‘Phi’).
Its unit is the weber with the unit symbol $\mathrm{Wb}$.

$$[\Phi ]=\mathrm{Wb}=\mathrm{V}\mathrm{s}=\mathrm{T}{\mathrm{m}}^2$$

2 Practical and worked examples

Magnetic flux is the surface integral of the magnetic flux density over the magnetically effective area:

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

2.1 Homogeneous field over a cross-section

For a homogeneous field and a constant cross-section, the following applies:

$$\Phi =B\cdot A$$

Example: $B=0.20\mathrm{T}$, $A=1.0\cdot 10^{-4}{\mathrm{m}}^2$

$$\Phi =0.20\cdot 1.0\cdot 10^{-4}\mathrm{T}{\mathrm{m}}^2=2.0\cdot 10^{-5}\mathrm{Wb}=20\mu \mathrm{Wb}$$

2.2 Magnetic circuit (reluctance model)

With the reluctance $R_{\mathrm{m}}=\frac{l}{\mu A}$ and the magnetomotive force $\Theta =N\cdot I$, the following applies:

$$\Phi =\frac{\Theta }{R_{\mathrm{m}}}$$

Example: $\Theta =100\mathrm{A}$, $R_{\mathrm{m}}=2.0\cdot 10^5\frac{\mathrm{A}}{\mathrm{Wb}}$

$$\Phi =\frac{100}{2.0\cdot 10^5}\mathrm{Wb}=5.0\cdot 10^{-4}\mathrm{Wb}=0.50\mathrm{mWb}$$

2.3 Electromagnetic induction (Faraday)

The induced voltage $u(t)$ of an inductor with $N$ turns is:

$$u(t)=-N\frac{\mathrm{d}\Phi }{\mathrm{d}t}=-\frac{\mathrm{d}}{\mathrm{d}t}(N\Phi )=-\frac{\mathrm{d}\Lambda }{\mathrm{d}t}$$

Example: $\Phi$ decreases linearly from $20\mu \mathrm{Wb}$ to $0$ within $2.0\mathrm{ms}$, $N=200$

$$\frac{\mathrm{d}\Phi }{\mathrm{d}t}\approx \frac{-20\cdot 10^{-6}}{2.0\cdot 10^{-3}}\frac{\mathrm{Wb}}{\mathrm{s}}=-1.0\cdot 10^{-2}\frac{\mathrm{Wb}}{\mathrm{s}}$$ $$u=-N\frac{\mathrm{d}\Phi }{\mathrm{d}t}=-200\cdot (-1.0\cdot 10^{-2})\frac{\mathrm{V}\mathrm{s}}{\mathrm{s}}\approx 2.0\mathrm{V}$$

3 Typical orders of magnitude

• small relay/sensor inductors: $\Phi \approx 1\dots 100\mu \mathrm{Wb}$
• small transformers/electromagnets: $\Phi \approx 0.1\dots 5\mathrm{mWb}$
• large machines/electromagnets: $\Phi \approx 0.01\dots 1\mathrm{Wb}$

4 Example phrasings

• ‘The magnetic flux through the core is $0.8\mathrm{mWb}$.’
• ‘In the air gap, a flux of $25\mu \mathrm{Wb}$ flows.’
• ‘The change in flux produces $u=-N\mathrm{d}\Phi /\mathrm{d}t$.’

5 Related quantities and relationships

• Magnetic flux is the surface integral of the magnetic flux density over the magnetically effective area:
  $\Phi ={\int }_A\vec{B}\cdot \mathrm{d}\vec{A}$
• Homogeneous cross-section:
  $B=\frac{\Phi }{A}$
• Magnetomotive force and reluctance:
  $\Phi =\frac{\Theta }{R_{\mathrm{m}}},R_{\mathrm{m}}=\frac{l}{\mu A}$
• Flux linkage and inductance:
  $\Lambda =N\Phi ,L=\frac{\Lambda }{I}=\frac{N\Phi }{I}\Rightarrow L=\frac{N^2}{R_{\mathrm{m}}}\text{(linear magnetic circuit)}$
• Law of induction (time-varying flux):
  $u(t)=-N\frac{\mathrm{d}\Phi }{\mathrm{d}t}$

6 Related components

• Inductor, transformer
• Relay and contactor