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

Electrostatics

01Coulomb's Law: Force Between Two Point Charges 02Superposition: Adding Coulomb Forces as Vectors 03Field Lines, Null Points and Uniform Fields 04The Electric Dipole and Its 1/r³ Field 05Torque, Energy and Force on a Dipole in a Field Problem set0/10 Problem set 20/10 MIT problem set0/1

01Charge Density and the Field on a Rod's Axis 02The Field on a Charged Rod's Perpendicular Bisector 03The Fields of a Charged Ring and a Charged Disk 04How Field Lines Carry Force, Energy and Momentum 05Electric Pressure and Tension: The Stress ε₀E²/2 Problem set0/10 Problem set 20/10 MIT problem set0/6

The Fields of a Charged Ring and a Charged Disk

How does stacking rings build a charged disk's field, and why does it jump by σ/ε₀ across the surface?

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Your summary note

1. Ring on its axis: $E_z = \frac{1}{4\pi\varepsilon_0}\frac{Qz}{(R^2+z^2)^{3/2}}$
  Write $dq = \lambda R\,d\phi'$ , pair opposite elements so only $dE\cos\theta$ survives, integrate over $\phi'$ , and record that $E_z$ vanishes at the centre and peaks at $z = R/\sqrt{2}$ .
2. Disk built from rings: $E_z = \frac{\sigma}{2\varepsilon_0}\left[1 - \frac{z}{\sqrt{z^2+R^2}}\right]$
  Give the ring of radius $r'$ the charge $\sigma\,2\pi r'\,dr'$ , insert it into the ring result, integrate with $u = r'^2+z^2$ , and write the $z>0$ and $z<0$ cases.
3. The limits $z \gg R$ and $R \gg z$
  Taylor-expand $(1+R^2/z^2)^{-1/2}$ to recover $\frac{1}{4\pi\varepsilon_0}\frac{Q}{z^2}$ with $Q = \sigma\pi R^2$ , then write the infinite-plane field $\pm\frac{\sigma}{2\varepsilon_0}\hat{\mathbf{k}}$ and sketch its step graph.
4. Jump in the normal field, $\Delta E_n = \sigma/\varepsilon_0$
  Compute $E_{z+} - E_{z-}$ across the plane from the two $\pm\frac{\sigma}{2\varepsilon_0}$ values, and state the general rule for the normal component across any surface carrying charge density $\sigma$ .
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