Length

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

Length (German: Länge) describes the spatial extent of something or the distance between two points. In electrical engineering, length occurs, among other things, as conductor length, plate separation, wavelength or path length.

1 Quantity symbol and unit

Common quantity symbols are

• $l$ (length of a component/conductor),
• $d$ (distance between capacitor plates, thickness of a plate or diameter of a round body, e.g. of a conductor or of an inductor core),
• $s$ (path/distance in kinematics).

The unit is the metre (m):

$$[l]=[d]=[s]=\mathrm{m}$$

Definition:
The metre is the length of the path travelled by light in vacuum during a time interval of $\frac{1}{299792458}\mathrm{s}$. Length is thus defined via the constant speed of light $c=299792458\frac{\mathrm{m}}{\mathrm{s}}$.

Useful conversions:

$$1\mathrm{m}=100\mathrm{cm}=1000\mathrm{mm},1\mathrm{km}=1000\mathrm{m}$$

2 Practical and worked examples

Conductor resistance (German: Leiterwiderstand) (effective length including outgoing and return conductors):

$$R=\rho \cdot \frac{l}{A}$$

Example: copper $\rho =0.0172\frac{\Omega \mathrm{m}{\mathrm{m}}^2}{\mathrm{m}}$, $A=1.5\mathrm{m}{\mathrm{m}}^2$, electric line $2\times 20\mathrm{m}\Rightarrow l=40\mathrm{m}$
$R=0.0172\cdot \frac{40}{1.5}\mathrm{\Omega }\approx \underline{0.46\mathrm{\Omega }}$
Voltage drop at $I=10\mathrm{A}$: $\Delta U=I\cdot R\approx \underline{4.6\mathrm{V}}$.

Wavelength (German: Wellenlänge) (electromagnetic wave in air):

$$\lambda =\frac{c}{f}$$

Example: $f=2.4\mathrm{GHz}\Rightarrow \lambda \approx \underline{0.125\mathrm{m}}$.

Signal propagation time in cables (phase velocity $v\approx (1.5\text{...}2.1)\cdot 10^{8}m/s$):

$$t=\frac{l}{v}$$

Example: $l=100\mathrm{m},v=2.0\cdot 10^8\frac{\mathrm{m}}{\mathrm{s}}\Rightarrow t=\underline{0.5\mu \mathrm{s}}$.

Plate separation in the capacitor (German: Plattenabstand im Kondensator) (influence of the length $d$):

$$C={\epsilon }_0{\epsilon }_r\frac{A}{d}$$

Air (${\epsilon }_r=1$): $A=25\mathrm{c}{\mathrm{m}}^2=2.5\cdot 10^{-3}{\mathrm{m}}^2$, $d=0.30\mathrm{mm}=3.0\cdot 10^{-4}\mathrm{m}$ $C=\frac{8.854\cdot 10^{-12}F/m\cdot 2.5\cdot 10^{-3}{\mathrm{m}}^2}{3.0\cdot 10^{-4}\mathrm{m}}\approx 7.38\cdot 10^{-11}\mathrm{F}=\underline{73.8\mathrm{pF}}$

With polytetrafluoroethylene (PTFE, brand name: Teflon) as the dielectric (${\epsilon }_r\approx 2.1$):

$$C\approx 2.1\cdot 73.8\mathrm{pF}\approx \underline{155\mathrm{pF}}$$

3 Typical orders of magnitude

• Separation of capacitor plates: $\mathrm{nm}\dots \mu \mathrm{m}$
• Device/cable lengths: $\mathrm{cm}\dots \mathrm{m}$
• Line sections/routes: $\mathrm{km}$

4 Example phrasings

• ‘The electric line has a length of $12\mathrm{m}$.’
• ‘The plate separation is $2\mathrm{mm}$.’
• ‘The antenna length is matched to the wavelength $\lambda$.’

5 Related quantities

• Velocity
• Wavelength
• Conductor resistance

6 Related components

• Conductor, electric line, cable
• Capacitor (plate separation)
• Inductor
• Transformer (magnetic length)