Current density

Prof. Dr. Thorsten Jungmann (last updated 2025-09-03)

Current density (German: Stromdichte) describes how much current flows through a conductor cross-section per unit area. It is an important measure of the thermal load on conductors and determines their current-carrying capacity.

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 current density is $\vec{J}$ or $J$ (as a scalar quantity).
Its unit is ampere per square metre or ampere per square millimetre:

$$[J]=A/{\mathrm{m}}^2=A/m{\mathrm{m}}^2$$

2 Practical and worked examples

Current density is defined as the quotient of current and cross-sectional area:

$$J=\frac{I}{A}$$

2.1 Current density in a conductor

The current density in a conductor with a cross-section of $A=4\mathrm{m}{\mathrm{m}}^2$ at a current of $I=20\mathrm{A}$:

$$J=\frac{I}{A}=\frac{20\mathrm{A}}{4\mathrm{m}{\mathrm{m}}^2}=\underline{5A/m{\mathrm{m}}^2}$$

2.2 Domestic installation according to DIN VDE

Typical values for installation cables:

• $A=1.5\mathrm{m}{\mathrm{m}}^2$ for $I_{max}=16\mathrm{A}$: $J=\frac{16\mathrm{A}}{1.5\mathrm{m}{\mathrm{m}}^2}=\underline{10.7A/m{\mathrm{m}}^2}$
• $A=2.5\mathrm{m}{\mathrm{m}}^2$ for $I_{max}=20\mathrm{A}$: $J=\frac{20\mathrm{A}}{2.5\mathrm{m}{\mathrm{m}}^2}=\underline{8.0A/m{\mathrm{m}}^2}$
• $A=4.0\mathrm{m}{\mathrm{m}}^2$ for $I_{max}=25\mathrm{A}$: $J=\frac{25\mathrm{A}}{4.0\mathrm{m}{\mathrm{m}}^2}=\underline{6.25A/m{\mathrm{m}}^2}$

2.3 Determining the cross-section

Determine the required cross-section for a given current and permissible current density:
For $I=32\mathrm{A}$ and $J_{rated}=8A/m{\mathrm{m}}^2$:

$$A_{min}=\frac{I}{J_{rated}}=\frac{32\mathrm{A}}{8A/m{\mathrm{m}}^2}=\underline{4.0\mathrm{m}{\mathrm{m}}^2}$$

3 Typical orders of magnitude

• Domestic installation (permissible continuous load): $J\approx 6$ to $11A/m{\mathrm{m}}^2$
• Industrial busbars: $J\approx 1$ to $3A/m{\mathrm{m}}^2$
• Electric motors (rated operation): $J\approx 4$ to $8A/m{\mathrm{m}}^2$
• Transformer windings: $J\approx 2$ to $6A/m{\mathrm{m}}^2$
• Short-time operation (starting currents): $J\approx 15$ to $30A/m{\mathrm{m}}^2$

4 Example phrasings

• ‘The current density in the conductor is $8.0A/m{\mathrm{m}}^2$.’
• ‘A 16 A electric line with $1.5\mathrm{m}{\mathrm{m}}^2$ has a maximum current density of $10.7A/m{\mathrm{m}}^2$.’
• ‘For continuous operation at $25\mathrm{A}$, a cross-section of at least $4\mathrm{m}{\mathrm{m}}^2$ is required.’
• ‘The high current density during start-up of the electric motor is permissible only for a short time.’

5 Related quantities

• Current and cross-section:

$$J=\frac{I}{A}$$

• Maximum current-carrying capacity:

$$I_{max}=J_{rated}\cdot A$$

• Local form of Ohm’s law with conductivity:

$$\vec{J}=\sigma \cdot \vec{E}$$

• Power loss per volume:

$$p_{\mathrm{loss}}=J^2\cdot \rho =\frac{J^2}{\sigma }$$

6 Related components

• Conductor, electric line, cable: dimensioning according to the permissible current density
• Transformer: winding cross-sections are designed according to the current density.
• Electric motor: windings must be dimensioned thermally.