EM fields lecture 26 - direction cosines, multipole coefficients & moments and point charge examples

Overview

Tim Gorringe explains how direction cosines, multipole coefficients, and Cartesian moments describe the electrostatic potential of charge distributions. He works through a point charge at the origin and the same charge displaced to z = A, showing how the monopole remains Q while dipole and quadrupole moments change with the coordinate origin, even though the complete potential remains unchanged.

Key takeaways

Direction cosines Lx, Ly, and Lz encode the field-point orientation, while the explicit factors 1/R, 1/R², and 1/R³ encode the radial falloff of monopole, dipole, and quadrupole contributions.
Cartesian multipole moments separate charge-position information from field-point direction: p_j = Σᵢ qᵢxᵢⱼ and Q_jk = Σᵢ qᵢ(3xᵢⱼxᵢₖ − rᵢ²δ_jk).
A point charge Q at the origin has monopole moment Q but zero dipole and quadrupole moments, so its potential is exactly Q/(4πε₀R).
Moving that charge to z = A preserves the monopole Q but produces pz = QA and quadrupole components Qzz = 2QA² and Qxx = Qyy = −QA².
When the total charge is nonzero, the monopole is the lowest nonzero moment and is invariant under an origin change, while higher moments generally depend on the origin.
Origin-dependent higher multipole moments do not make the physical result ambiguous: the complete electrostatic potential remains invariant when all contributions are combined.

Chapters

0:00 From Multipole Expansion Theory to Point-Charge Examples

The lecture applies the multipole expansion to point-charge arrangements, progressing from monopole behavior to dipole and quadrupole terms.
Direction cosines and a comparison of multipole coefficients with multipole moments provide the tools for the worked examples.

1:52 Direction Cosines Encode the Field-Point Direction

The three direction cosines, Lx, Ly, and Lz, are the Cartesian components of the unit vector pointing toward the field point.
In Cartesian coordinates they are x/R, y/R, and z/R; with the standard spherical convention they are sinθ cosφ, sinθ sinφ, and cosθ.
Direction cosines describe orientation only, while R separately represents the distance to the field point.

7:27 Separating Radial Dependence from Multipole Coefficients

The potential expansion pulls out its radial factors: 1/R for the monopole, 1/R² for the dipole, and 1/R³ for the quadrupole.
Each multipole coefficient depends on the field-point direction and the charge arrangement, but not on the field-point distance.
Multipole moments further isolate information about charge positions, independent of the direction toward the field point.

13:00 Legendre Coefficients for Monopole, Dipole, and Quadrupole Terms

The monopole, dipole, and quadrupole correspond to expansion orders L = 0, 1, and 2, with Legendre polynomials P0, P1, and P2.
Their potential contributions have successive inverse powers of R, making the radial falloff distinguish each order.
The same pattern extends to the octupole at L = 3 and higher multipole terms.

16:00 Cartesian Dipole and Quadrupole Moments

The monopole moment is the total charge, while the dipole coefficient is the direction-cosine-weighted sum L_j p_j.
The dipole moment has three components, p_j = Σᵢ qᵢ xᵢⱼ, determined by each charge and its Cartesian position.
The quadrupole coefficient is Σⱼₖ L_j L_k Q_jk, where Q_jk = Σᵢ qᵢ(3xᵢⱼxᵢₖ − rᵢ²δ_jk).
The quadrupole moment is a 3 by 3 rank-two tensor; the Kronecker delta contributes the rᵢ² subtraction only to diagonal elements.

25:30 A Point Charge at the Origin Has Only a Monopole Moment

For one charge Q at the origin, the monopole moment is Q because it is the sum of all charges.
Every dipole component vanishes because the charge's x, y, and z coordinates are zero.
All quadrupole components also vanish, leaving V(R) = Q/(4πε₀R), the familiar point-charge potential.

34:30 Moving the Charge to z = A Creates Dipole and Quadrupole Moments

Displacing the charge from the origin to z = A leaves the monopole moment unchanged at Q.
The dipole moment becomes pz = QA, while px and py remain zero.
The quadrupole diagonal becomes Qzz = 2QA² and Qxx = Qyy = −QA²; off-diagonal components vanish.
The example shows that moving the origin relative to a charge can make higher-order moments nonzero.

43:00 The Lowest Nonzero Moment Is Intrinsic, Higher Moments Depend on Origin

For a distribution with nonzero total charge, the monopole moment is the lowest nonzero moment and remains fixed when the coordinate origin changes.
Dipole, quadrupole, and higher moments can change with the chosen origin, as the single-charge examples demonstrate.
Changing the origin redistributes contributions among higher-order terms rather than changing the underlying charge distribution.

47:00 Taylor-Series Analogy and Origin-Independent Potential

The origin dependence of higher multipole moments is compared with Taylor coefficients, whose higher derivatives depend on the point about which a function is expanded.
Although individual higher-order terms vary with the coordinate origin, the full electrostatic potential remains unique.
The multipole expansion is useful because its complete sum describes the same physical potential regardless of how the origin changes the mix of moments.

Transcript excerpt

0:01 Last class we talked about the multipole expansion. We started our work on the multipole expansion. Specifically, we talked about the concept or idea of a multipole expansion, and we derived the equation formula for the multipole expansion. Uh today's class, I mostly want to work on examples and illustration of the multipole expansion. So, we're going to look at applying it to uh some charge distributions, some arrangements of point charges, from the most simple arrangement of point charges uh to more complicated arrangements where we expose the dipole term, the quadrupole term, and so

0:50 forth. Uh before I do that, I just wanted to cover uh direction cosines, which are one way that we can write down the multipole expansion. Uh I did mention them at the end of the last class, but I want to talk about them a little more at the beginning of this class. Uh I also wanted to just walk through uh by making a table of the terms in the multipole expansion in the way that we the possible ways we can describe uh a charge distribution uh in the context of the multipole expansion and the resulting potential of that charge distribution. So, that that is set us going on the examples. The

1:37 direction cosines. Um I will use the symbol L but lowercase L for direction cosines. They use that symbol in the uh in the book. Um there are three direction cosines. Uh LX, LY, LZ. Uh so I'll uh often use a label J or um a label K uh to denote the uh different direction cosines. Sometimes we need uh just one direction cosine, so I LJ is

2:25 sufficient. Um sometimes we need a product of them and we might use LJ and LK. Uh so the direction cosines

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Related notes

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, tim gorringe.