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

Closed Surface Integrals: The Sphere and Its Charge

How does the spherical area element sum to 4πR², and how do you integrate over a closed surface built from pieces?

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Your summary note

1. Closed surfaces and the circled double-integral sign
  Define a closed surface as one that completely encloses a volume, give two examples, and record that the loop on $\oint\!\!\oint$ marks closure as it does for closed paths.
2. Sphere area $A = 4\pi R^2$ and its area element
  Write the sphere's area as a closed surface integral, write $A = 4\pi R^2$ with $\theta$ polar and $\phi$ azimuthal, and note that math texts sometimes swap the two angles.
3. Uniform charge on a sphere, $Q = \sigma \, 4\pi R^2$
  Write $Q$ as the closed surface integral of $\sigma \, dA$ , pull the constant $\sigma$ out, replace the remaining integral by the known area, and check the units come out in coulombs.
4. Total charge on the two-hemisphere-and-annulus surface
  Sketch the surface, split the closed integral into three open-surface integrals with one density each, then use hemisphere area $2\pi R^2$ and annulus area $\pi(R_2^2 - R_1^2)$ to reach $Q$ .
Attempt 1 of 2