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 Field on a Charged Rod's Perpendicular Bisector

Why does symmetry cancel the sideways field beside a charged rod, and how does a tan substitution finish the integral?

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Your summary note

1. Symmetry cancels $E_x$ , leaving $dE_y = dE\cos\theta'$
  Write $dq = \lambda\, dx'$ and $dE$ at distance $\sqrt{x'^2 + y^2}$ , sketch paired elements at $\pm x'$ with their contributions at $P$ , and reduce $dE_y$ to $P$ .
2. The substitution $x' = y\tan\theta'$ and the result
  Integrate $dE_y$ from $-\ell/2$ to $\ell/2$ , carry the substitution and new limits $\pm\theta$ through to $\frac{2\sin\theta}{y^2}$ , and write $E_y = \frac{1}{4\pi\varepsilon_0}\frac{2\lambda\sin\theta}{y}$ in terms of $\ell$ and $y$ .
3. The limits $y \gg \ell$ and $\ell \gg y$
  Take each limit of the result, showing $y \gg \ell$ gives the point-charge field $\frac{Q}{4\pi\varepsilon_0 y^2}$ and $\ell \gg y$ gives the infinite-line field $\frac{2\lambda}{4\pi\varepsilon_0 y}$ , with units checked.
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