ELC-31

Maxwell displacement current

Id = ε A dE/dt. The missing current that closes Ampère in a charging capacitor.

Reading speed

FieldsMaxwell 1861

Governing equation

I_d=\varepsilon A\dfrac{\mathrm{d}E}{\mathrm{d}t}

$\mathrm{d}E/\mathrm{d}t \in \left[0.01, 80\right]$

where

\varepsilon_r: Relative ε (—)
A: Plate area (cm^2)
\mathrm{d}E/\mathrm{d}t: E slope (MV/m/s)
I_d: Displacement current (µA)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

I_d=\varepsilon A\dfrac{\mathrm{d}E}{\mathrm{d}t}

$\mathrm{d}E/\mathrm{d}t \in \left[0.01, 80\right]$

$\mathrm{d}E/\mathrm{d}t = 0.01$ $I_d = 3.54\times 10^{-4}$

Along the curve

②

Domain of the variable — blue

\mathrm{d}E/\mathrm{d}t_{\min} = 0.01,\quad \mathrm{d}E/\mathrm{d}t_{\max} = 80

$\mathrm{d}E/\mathrm{d}t \in \left[0.01, 80\right]$

$\mathrm{d}E/\mathrm{d}t = 0.01$ $\mathrm{d}E/\mathrm{d}t = 0.01$

Along the curve

③

Response curve — green

I_d=\varepsilon A\dfrac{\mathrm{d}E}{\mathrm{d}t}

$\mathrm{d}E/\mathrm{d}t \in \left[0.01, 80\right]$

$\mathrm{d}E/\mathrm{d}t = 0.01$ $I_d = 3.54\times 10^{-4}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}I_d}{\mathrm{d}\mathrm{d}E/\mathrm{d}t}

$\mathrm{d}E/\mathrm{d}t \in \left[0.01, 80\right]$

$\mathrm{d}E/\mathrm{d}t = 0.513$ $I_d^{\prime} = 0.035$

Along the curve

Lecture brief

Historical brief

Ohm (1827), Kirchhoff (1845) and Maxwell’s circuit reduction still run every board: RLC transients, transformers, skin effect and three-phase power. The sheets are those network laws, not a SPICE deck. This sheet (ELC-31 — Maxwell displacement current) is the form associated with Maxwell 1861. Working symbols: $\varepsilon_r$ , $A$ , $\mathrm{d}E/\mathrm{d}t$ $\rightarrow$ $I_d$ . Maxwell added ∂D/∂t to Ampère so that charge is conserved and electromagnetic waves exist.

Purpose

Purpose: compute

I_d

from

\varepsilon_r

A

\mathrm{d}E/\mathrm{d}t

in Electrical via

I_d=\varepsilon A\dfrac{\mathrm{d}E}{\mathrm{d}t}

Id = ε A dE/dt. The missing current that closes Ampère in a charging capacitor. Use it when a real electrical question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\varepsilon_r = 1.000\,\mathrm{—}

A = 40.000\,\mathrm{cm^2}

\mathrm{d}E/\mathrm{d}t = 5.000\,\mathrm{MV/m/s}

, the governing relation

I_d=\varepsilon A\dfrac{\mathrm{d}E}{\mathrm{d}t}

yields

I_d = 0.177\,\mathrm{\mu A}

. A charging capacitor, an Id arrow between plates. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Relative ε \varepsilon_r1 —Plate area A40 cm^2E slope \mathrm{d}E/\mathrm{d}t5 MV/m/s

Outputs

Displacement current I_d0.177 µA

Reading speed

Watch on YouTube

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Narration of this film

A charging capacitor, an Id arrow between plates.

Maxwell added ∂D/∂t to Ampère so that charge is conserved and electromagnetic waves exist.

Reading speed

Watch on YouTube