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

The Electric Dipole and Its 1/r³ Field

Why does a dipole's field fall off as 1/r³ instead of 1/r², and which way does it point far away?

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Your summary note

1. Dipole moment $\vec{p} = 2qa\,\hat{j}$
  Define $\vec{p}$ for $\pm q$ separated by $2a$ , pointing from $-q$ to $+q$ , give $\vec{p} = \sum_i q_i \vec{r}_i$ for a neutral system, and name polar molecules and induced dipoles as examples.
2. Exact field components from two point charges
  Superpose the fields of $+q$ at $y = a$ and $-q$ at $y = -a$ to write $E_x$ and $E_y$ at $P(x, y, 0)$ in terms of $x$ , $y$ and $a$ .
3. The point-dipole limit $r \gg a$
  Reduce the exact components to $E_x = \frac{3p\sin\theta\cos\theta}{4\pi\varepsilon_0 r^3}$ and $E_y = \frac{p(3\cos^2\theta - 1)}{4\pi\varepsilon_0 r^3}$ , then assemble the vector form using $\vec{p} \cdot \vec{r}$ .
4. Inverse-cube falloff and the two field-line pictures
  Contrast the dipole's $1/r^3$ falloff with a point charge's $1/r^2$ , tie it to the dipole's zero net charge, and sketch the finite-dipole and point-dipole field lines side by side.
Attempt 1 of 2