Total Charge on a Plate

Charge spreads unevenly over a plate. Chop the plate into small squares, multiply each square's charge density by its area, and add: as the squares shrink, that sum becomes a double integral.

Drag on the surface to move the point, or click a red marker to jump to it.

How it works

Double integrals

In one sentence

The double integral $\iint_R \sigma\,dA$ is the limit of Riemann sums $\sum \sigma(x_i, y_j)\,\Delta A$ as the rectangles shrink; for a continuous $\sigma$ on a rectangle it equals either iterated integral (Fubini's theorem).

General formula

{\htmlData{sym=0}{Q}} = \iint_{\htmlData{sym=4}{R}} {\htmlData{sym=1}{\sigma}}(x, y)\,{\htmlData{sym=2}{dA}} \;\approx\; \sum_{i,j}{\htmlData{sym=1}{\sigma}}(x_i, y_j)\,{\htmlData{sym=3}{\Delta x\,\Delta y}}

Each term is density times area, which is charge, so the sum is total charge. Shrinking the rectangles removes the error from treating $\sigma$ as constant on each one.

What the symbols mean

$Q$
Total charge
All the charge on the plate, in nanocoulombs.
$\sigma$
Charge density
Charge per square meter at each point; the height of the surface.
$dA$
Area element
A tiny piece of the plate's area.
$\Delta x\,\Delta y$
Box footprint
The area of one rectangle under a box.
$R$
Region
The plate itself: the rectangle 0 ≤ x ≤ 2, 0 ≤ y ≤ 1.

Each box stands on one small rectangle of area $\Delta A = \Delta x\,\Delta y$ , with height equal to the density $\sigma$ at its center, so its volume is that rectangle's charge.
Adding the boxes gives the Riemann sum; more, thinner boxes hug the surface more closely.
In the limit the sum becomes $\iint_R \sigma\,dA$ , the volume under the surface: the total charge.
Iterated integrals compute it one direction at a time: $\int_0^2\!\int_0^1 (a + bxy)\,dy\,dx = 2a + b$ .

Learn the math

Double IntegralsCalc III · Lesson 30, Multiple Integrals Iterated IntegralsCalc III · Lesson 31, Multiple Integrals Fields of Continuous ChargeE&M · Lesson 3, Charge and Electric Fields Conductors and ShieldingE&M · Lesson 7, Gauss's Law Maxwell's EquationsE&M · Lesson 27, Maxwell's Equations and Light

Learn the math

Double IntegralsCalc III · Lesson 30, Multiple Integrals

Iterated IntegralsCalc III · Lesson 31, Multiple Integrals

Fields of Continuous ChargeE&M · Lesson 3, Charge and Electric Fields

Conductors and ShieldingE&M · Lesson 7, Gauss's Law

Maxwell's EquationsE&M · Lesson 27, Maxwell's Equations and Light