EMG-10

Parallel-plate capacitor

C = ε0 εr A / d, E = V/d.

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CapacitanceCavendish / Maxwell

Governing equation

C=\varepsilon_0\varepsilon_r\dfrac{A}{d},\quad E=\dfrac{V}{d}

$d \in \left[0.1, 20\right]$

where

A: Area (cm²)
d: Gap (mm)
\varepsilon_r: Relative permittivity (—)
V: Voltage (V)
C: Capacitance (pF)
E: Field (kV/m)

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

C=\varepsilon_0\varepsilon_r\dfrac{A}{d},\quad E=\dfrac{V}{d}

$d \in \left[0.1, 20\right]$

$d = 0.1$ $C = 885.4$

Along the curve

②

Domain of the variable — blue

d_{\min} = 0.1,\quad d_{\max} = 20

$d \in \left[0.1, 20\right]$

$d = 0.1$ $d = 0.1$

Along the curve

③

Response curve — green

C=\varepsilon_0\varepsilon_r\dfrac{A}{d},\quad E=\dfrac{V}{d}

$d \in \left[0.1, 20\right]$

$d = 0.1$ $C = 885.4$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}C}{\mathrm{d}d}

$d \in \left[0.1, 20\right]$

$d = 0.225$ $C^{\prime} = -2527.5$

Along the curve

Lecture brief

Historical brief

Coulomb, Gauss, Ampère, Faraday and Maxwell (1861–65) unified charge, current and light. The lab computes fields, induction, Poynting flux and the electromagnetic wave in SI. This sheet (EMG-10 — Parallel-plate capacitor) is the form associated with Cavendish / Maxwell. Working symbols: $A$ , $d$ , $\varepsilon_r$ , $V$ $\rightarrow$ $C$ , $E$ . Gauss's law between infinite plates gives σ/ε, so C = Q/V = ε A/d. A dielectric multiplies ε0 by εr.

Purpose

Purpose: compute

C

E

from

A

d

\varepsilon_r

V

in Electromagnetism via

C=\varepsilon_0\varepsilon_r\dfrac{A}{d},\quad E=\dfrac{V}{d}

C = ε0 εr A / d, E = V/d. Use it when a real electromagnetism question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

A = 100.000\,\mathrm{cm^{2}}

d = 1.000\,\mathrm{mm}

\varepsilon_r = 1.000\,\mathrm{—}

V = 12.000\,\mathrm{V}

, the governing relation

C=\varepsilon_0\varepsilon_r\dfrac{A}{d},\quad E=\dfrac{V}{d}

yields

C = 88.542\,\mathrm{pF}

E = 12.000\,\mathrm{kV/m}

. Fringing neglected, linear dielectric. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Area A100 cm²Gap d1 mmRelative permittivity \varepsilon_r1 —Voltage V12 V

Outputs

Capacitance C88.542 pF
Field E12.000 kV/m

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Narration of this film

Fringing neglected, linear dielectric.

Gauss's law between infinite plates gives σ/ε, so C = Q/V = ε A/d. A dielectric multiplies ε0 by εr.

Reading speed

Watch on YouTube