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

Electric Flux Through Surfaces and Volume Integrals

How does E · dA pick out the part of a field piercing a surface, and how do triple integrals extend the sum to volumes?

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Your summary note

1. The area vector $d\mathbf{A}$ and $\mathbf{E} \cdot d\mathbf{A} = E_n \, dA$
  Define $\Phi_E = \iint \mathbf{E} \cdot d\mathbf{A}$ , state the direction rule for $d\mathbf{A}$ on closed and open surfaces, and rewrite the integrand as $E_n \, dA$ with $E_n$ the normal component.
2. Flux of a point charge's field through a sphere
  Work the sphere of radius $R$ centred on the charge, pulling the constant $E_n$ out to get $\Phi_E = E_n \, 4\pi R^2$ , and record the strategy of splitting a closed surface into pieces where $E_n = 0$ or $E_n$ is constant.
3. Triple integrals over a box: volume and charge
  Write $V = \iiint dx\,dy\,dz$ with limits for a box of sides $a$ , $b$ , $c$ , integrate from the inside out to get $abc$ , then set up $Q = \iiint \rho \, dV$ and evaluate it for uniform $\rho_0$ .
4. The spherical volume element $dV = r^2 \sin\theta \, dr\,d\theta\,d\phi$
  State the volume element in spherical coordinates and record the physics convention that $\theta$ is the polar angle and $\phi$ the azimuthal angle.
Attempt 1 of 2