Electricity and Magnetism

Mathematical Tools and Fields

01Series Approximations: Keeping One or Two Terms 02Integrals as Sums: Along a Line and a Curved Path 03Line Integrals: Work as the Integral of F · ds 04Closed Line Integrals: Work Around a Loop 05Double Integrals: Area, Charge and Volume 06Closed Surface Integrals: The Sphere and Its Charge 07Electric Flux Through Surfaces and Volume Integrals Problem set0/10 Problem set 20/10 MIT problem set0/4

01Field Theory Versus Action at a Distance 02Scalar Fields: Contour, Color and Relief Maps 03Vector Fields, Sources, Sinks and Divergence 04Curl: Measuring the Circulation of a Field 05Gravitational, Electric and Magnetic Fields 06Drawing Vector Fields: Arrows, Lines and Grass Seeds Problem set0/10 Problem set 20/10 MIT problem set0/1

Double Integrals: Area, Charge and Volume

How do small area elements dx dy and r dr dθ add up to an area, a total charge, or a volume?

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Your summary note

1. The area element $dA = dx\,dy$ and the double integral
  Write the double integral as the limit of $\sum_i \sum_j \Delta x_i\,\Delta y_j$ , then evaluate the rectangle's area as two nested integrals to get $ab$ , noting what each integration sums.
2. Total charge from a surface density $\sigma(x,y)$
  Write $dQ = \sigma\,dx\,dy$ in C/m² times m², then integrate $\sigma = kxy$ over the $a \times b$ rectangle, holding $y$ fixed in the inner integral, to reach $Q = ka^2b^2/4$ .
3. Polar area element $dA = r\,d\theta\,dr$
  Sketch the patch with sides $dr$ and $r\,d\theta$ , then integrate $\sigma = kr^2$ over a disk of radius $R$ to obtain $Q = 2\pi k R^4/4$ .
4. Reducing to a single integral by circular symmetry
  Write the ring element $dA = 2\pi r\,dr$ , redo the disk charge as one integral over $r$ , and record that the shortcut fails once $\sigma$ depends on $\theta$ .
Attempt 1 of 2