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

Torque, Energy and Force on a Dipole in a Field

Why does a dipole in a uniform field feel a torque but no net force, and what changes when the field is non-uniform?

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Your summary note

1. Torque $\vec{\tau} = \vec{p} \times \vec{E}$ in a uniform field
  Show the net force vanishes, compute $\vec{r}_+ \times \vec{F}_+ + \vec{r}_- \times \vec{F}_-$ about the midpoint to get $\tau = pE\sin\theta$ along $-\hat{k}$ , and record the rotation of $\vec{p}$ toward $\vec{E}$ .
2. Potential energy $U = -\vec{p} \cdot \vec{E}$
  Integrate $dW = -pE\sin\theta\,d\theta$ from $\theta_0$ to $\theta$ , take $\Delta U = -W$ , choose $U(\pi/2) = 0$ , and reach $U(\theta) = -pE\cos\theta$ .
3. Stable and unstable orientations
  Identify the stable minimum $U_{\min} = -pE$ with $\vec{p}$ parallel to $\vec{E}$ and the unstable maximum $U_{\max} = +pE$ with $\vec{p}$ anti-parallel.
4. Net force $\vec{p}$ in a non-uniform field
  Expand the field to first order about the dipole's center to derive $\vec{F}_e = p\,(dE/dx)\,\hat{i}$ , and record the charged comb picking up bits of paper through induced dipoles as the example.
Attempt 1 of 2