The Multipole Expansion
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You should be comfortable with Legendre polynomials \( P_n(x) \) and the addition theorem for spherical harmonics, as well as basic vector calculus identities involving the gradient and divergence in spherical coordinates.
Why this matters
The multipole expansion allows us to approximate complex charge distributions with simple, calculable terms.
In electrostatics, calculating the potential of an arbitrary charge distribution directly is often impossible. However, for points far from the source, the potential can be expressed as a series of terms that decrease in magnitude. This expansion separates the problem into a "monopole" term (total charge), a "dipole" term (separation of charge), and higher-order "multipole" terms. This hierarchy is fundamental because it tells us which physical features of the charge distribution dominate the field at a distance, providing a systematic way to solve boundary value problems and understand field behavior without solving Laplace's equation from scratch for every geometry.
The Idea in Plain Language
Imagine looking at a complex object from a great distance. At first, you only see its total mass or charge; this is the monopole term. If you look closer, or if the object has no net charge, you notice it has a "head" and a "tail" of opposite signs; this is the dipole term. If the object is symmetric but has a more complex shape, like a barbell or a ring, you need the quadrupole term. The multipole expansion is simply a way of describing the electric field by peeling off these layers of complexity one by one. The key insight is that each successive term is smaller than the last by a factor related to the ratio of the source size to the distance from the observer. Therefore, for distant fields, the first few terms provide an excellent approximation, and the rest can often be ignored.
The Monopole and Dipole Terms
The expansion begins with the \( n=0 \) term, which depends only on the total charge \( Q \) of the distribution. This term is identical to the potential of a point charge located at the origin. It represents the leading-order behavior of the field and decays as \( 1/r \). If the total charge \( Q \) is non-zero, this term dominates the potential at large distances, and the field appears radial, as if all the charge were concentrated at the center.
The next term, for \( n=1 \), is the dipole term. It depends on the electric dipole moment \( \vec p \), defined as \( \vec p = \int \vec r' \rho(\vec r') \, d^3r' \). This term decays faster, as \( 1/r^2 \). It captures the first-order asymmetry of the charge distribution. For a system of point charges, the dipole moment is the sum of \( q_i \vec r_i \). Note that the dipole moment is independent of the choice of origin only if the total charge \( Q \) is zero. If \( Q \neq 0 \), the dipole moment depends on where you place the origin, which is why the monopole term is always the primary descriptor for charged systems.
Higher-Order Multipoles
The \( n=2 \) term is the quadrupole term. It decays as \( 1/r^3 \) and describes the deviation of the charge distribution from spherical symmetry beyond the dipole level. The quadrupole moment is a tensor \( Q_{ij} \), which is traceless and symmetric. For a linear charge distribution along the z-axis, the quadrupole moment is often denoted by \( Q \) (or \( Q_{zz} \)), and the potential term involves \( P_2(\cos\theta) = \frac{1}{2}(3\cos^2\theta - 1) \). Each subsequent term \( n \) decays as \( 1/r^{n+1} \) and involves the Legendre polynomial \( P_n(\cos\alpha) \), where \( \alpha \) is the angle between \( \vec r \) and \( \vec r^{\,\prime} \). The rapid decay of these higher-order terms means that for most practical purposes, the monopole and dipole terms are sufficient to describe the external field, unless the lower-order moments vanish due to symmetry.
The general term in the expansion is given by the integral involving the Legendre polynomial. The orthogonality of Legendre polynomials ensures that each term captures a distinct angular feature of the charge distribution. This mathematical structure allows us to isolate specific physical properties of the source, such as the shape of the charge cloud, without needing to know the exact position of every individual charge.
Convergence and Validity
The multipole expansion is a convergent series that converges absolutely for points outside the region containing the charge distribution. This means that if the source is bounded within a sphere of radius \( R \), the expansion is valid for all \( r > R \). The rate of convergence is determined by the ratio \( R/r \). As \( r \) increases, the higher-order terms become negligible very quickly. This makes the expansion a powerful tool for calculating fields in regions far from the source, such as in antenna theory or molecular physics, where the far field of a molecule is described by its dipole and quadrupole moments.
It is crucial to remember that this expansion is only valid outside the source. Inside the charge distribution, the potential cannot be expressed in this simple series form because the assumption \( r > r' \) fails for some source points. In such cases, one must use the full integral form or solve Laplace's equation with appropriate boundary conditions. The expansion provides a bridge between the local details of the charge distribution and the global behavior of the field, linking the microscopic structure to the macroscopic observables.
The derivation
Multipole Expansion of the Potential
For a static charge distribution \(\rho(\vec r')\) of bounded (compact) support, the electrostatic potential at a field point \(\vec r\) lying outside the source expands as an absolutely convergent series in inverse powers of \(r=|\vec r|\), \[ V(\vec r)=\frac{1}{4\pi\epsilon_0}\sum_{n=0}^{\infty}\frac{1}{r^{\,n+1}}\int (r')^{n}\,P_n(\cos\theta')\,\rho(\vec r')\,d^3r', \] whose first three terms are the monopole (\(\propto r^{-1}\)), dipole (\(\propto r^{-2}\)) and quadrupole (\(\propto r^{-3}\)) contributions, with \(\theta'\) the angle between \(\vec r\) and the source point \(\vec r'\), and \(P_n\) the Legendre polynomials.
A common pitfall is assuming the dipole moment is always a well-defined physical quantity independent of the coordinate origin. In reality, the dipole moment \( \vec p \) is only invariant under translations of the origin if the total charge \( Q \) is zero. If \( Q \neq 0 \), shifting the origin by \( \vec a \) changes the dipole moment by \( Q\vec a \). Therefore, for a net-charged system, the "dipole" term is not a unique property of the object but depends on your choice of reference point. Only the monopole term is truly origin-independent for a charged system. C
Worked example
Problem. A system consists of two point charges: \( q_1 = +2.0 \ \mathrm{\mu C} \) at \( \vec r_1 = (0, 0, 1.0 \ \mathrm{cm}) \) and \( q_2 = -2.0 \ \mathrm{\mu C} \) at \( \vec r_2 = (0, 0, -1.0 \ \mathrm{cm}) \). Calculate the electric potential at a point \( P \) located at \( \vec r = (0, 0, 10.0 \ \mathrm{cm}) \) using the first two terms of the multipole expansion (monopole and dipole). Compare this with the exact potential.
- Determine the total charge \( Q \) and the dipole moment \( \vec p \). \[ Q = q_1 + q_2 = 2.0 \ \mathrm{\mu C} - 2.0 \ \mathrm{\mu C} = 0 \] Since \( Q=0 \), the monopole term vanishes. The dipole moment is: \[ \vec p = q_1 \vec r_1 + q_2 \vec r_2 = (2.0 \times 10^{-6} \ \mathrm{C})(1.0 \times 10^{-2} \ \mathrm{m}) \hat z + (-2.0 \times 10^{-6} \ \mathrm{C})(-1.0 \times 10^{-2} \ \mathrm{m}) \hat z \] \[ \vec p = (2.0 \times 10^{-8} \ \mathrm{C \cdot m}) \hat z + (2.0 \times 10^{-8} \ \mathrm{C \cdot m}) \hat z = 4.0 \times 10^{-8} \ \mathrm{C \cdot m} \ \hat z \]
- Calculate the dipole potential at \( \vec r = 10.0 \ \mathrm{cm} \ \hat z \). Here, \( r = 0.10 \ \mathrm{m} \) and \( \hat r = \hat z \). \[ V_{\text{dip}} = \frac{1}{4\pi\epsilon_0} \frac{\hat r \cdot \vec p}{r^2} \] Using \( \frac{1}{4\pi\epsilon_0} = 8.99 \times 10^9 \ \mathrm{N \cdot m^2 / C^2} \): \[ V_{\text{dip}} = (8.99 \times 10^9) \frac{(4.0 \times 10^{-8})}{(0.10)^2} \] \[ V_{\text{dip}} = (8.99 \times 10^9) \frac{4.0 \times 10^{-8}}{0.01} = (8.99 \times 10^9) (4.0 \times 10^{-6}) \] \[ V_{\text{dip}} = 35.96 \ \mathrm{V} \]
- Calculate the exact potential for comparison. \[ V_{\text{exact}} = \frac{1}{4\pi\epsilon_0} \left( \frac{q_1}{r_1} + \frac{q_2}{r_2} \right) \] Distance to \( q_1 \): \( r_1 = 10.0 - 1.0 = 9.0 \ \mathrm{cm} = 0.09 \ \mathrm{m} \). Distance to \( q_2 \): \( r_2 = 10.0 - (-1.0) = 11.0 \ \mathrm{cm} = 0.11 \ \mathrm{m} \). \[ V_{\text{exact}} = (8.99 \times 10^9) \left( \frac{2.0 \times 10^{-6}}{0.09} - \frac{2.0 \times 10^{-6}}{0.11} \right) \] \[ V_{\text{exact}} = (8.99 \times 10^9) (2.0 \times 10^{-6}) \left( \frac{1}{0.09} - \frac{1}{0.11} \right) \] \[ \frac{1}{0.09} \approx 11.111, \quad \frac{1}{0.11} \approx 9.091 \] \[ \text{Difference} \approx 2.020 \] \[ V_{\text{exact}} = (17.98 \times 10^3) (2.020) \approx 36.32 \ \mathrm{V} \]
- Compare the results. The dipole approximation gives \( 35.96 \ \mathrm{V} \), while the exact value is \( 36.32 \ \mathrm{V} \). The relative error is approximately \( \frac{36.32 - 35.96}{36.32} \approx 1\% \). This small error confirms that the dipole term is a very good approximation at \( r = 10 \ \mathrm{cm} \), which is 10 times the distance of each charge from the origin (5 times their separation).
Units check. The dipole moment has units \( \mathrm{C \cdot m} \). The potential formula has \( \frac{\mathrm{N \cdot m^2 / C^2} \cdot \mathrm{C \cdot m}}{\mathrm{m^2}} = \frac{\mathrm{N \cdot m}}{\mathrm{C}} = \frac{\mathrm{J}}{\mathrm{C}} = \mathrm{V} \). The units are consistent.
Three quick tests
- If a charge distribution has a non-zero total charge, which term in the multipole expansion dominates at large distances?
Answer
The monopole term (\( 1/r \)) dominates because it decays the slowest. - Why is the dipole moment independent of the choice of origin only for neutral systems?
Answer
For a neutral system, the sum of charges is zero, so shifting the origin by \( \vec a \) adds \( \sum q_i \vec a = 0 \) to the dipole moment. For charged systems, it adds \( Q\vec a \). - At what distance \( r \) does the quadrupole term become comparable in magnitude to the dipole term, assuming a source size \( R \)?
Answer
Each successive order is smaller by roughly a factor \( R/r \), so the quadrupole term becomes comparable to the dipole term only when \( r \) is of the order of \( R \) — where the expansion is no longer useful.
Problem set for PU-102 →
Each derivation in this unit ends with five graded problems with full solutions.
Lecture notes drafted with a local model and checked line by line by the editor; every number in the worked example was recomputed.