6.6 Multipole expansion

Multipole expansion is a powerful technique for simplifying complex charge distributions. By breaking them down into simpler components like monopoles and dipoles, we can more easily calculate electric potentials and fields at large distances.

This method is crucial for understanding how electric charges interact over long ranges. It helps us analyze everything from molecular structures to the behavior of electric fields in space, making it a cornerstone of electromagnetic theory.

Multipole expansion overview

• Multipole expansion is a mathematical technique used to approximate the potential and field of a complex charge distribution at large distances
• Represents the charge distribution as a series of simpler charge configurations (monopole, dipole, quadrupole, etc.) with increasing complexity
• Allows for simplification of calculations in electrostatics and provides insights into the behavior of charge distributions

Electric potential of charge distributions

Potential in terms of charge density

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• Electric potential $V(\vec{r})$ at a point $\vec{r}$ due to a charge distribution with density $\rho(\vec{r}')$ is given by: $V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \int \frac{\rho(\vec{r}')}{|\vec{r} - \vec{r}'|} d^3r'$
• Charge density $\rho(\vec{r}')$ represents the amount of charge per unit volume at position $\vec{r}'$
• Integration is performed over the entire charge distribution

Convergence of multipole expansion

• Multipole expansion converges rapidly when the observation point $\vec{r}$ is far from the charge distribution compared to its size
• Convergence depends on the ratio of the charge distribution's size to the distance from the observation point
• Higher-order terms become increasingly less significant as the distance increases

Monopole moment

Net electric charge

• Monopole moment is the total electric charge $Q$ of the distribution
• Defined as the integral of the charge density over the entire volume: $Q = \int \rho(\vec{r}') d^3r'$
• Represents the zeroth-order term in the multipole expansion
• Dominant term when the observation point is far from the charge distribution

Dipole moment

Electric dipole definition

• Dipole moment $\vec{p}$ is a vector quantity that characterizes the separation of positive and negative charges in a charge distribution
• Defined as the product of the charge $q$ and the displacement vector $\vec{d}$ between the positive and negative charges: $\vec{p} = q\vec{d}$
• Measures the tendency of a charge distribution to align with an external electric field

Potential of electric dipole

• Electric potential due to a dipole at a point $\vec{r}$ is given by: $V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{\vec{p} \cdot \vec{r}}{r^3}$
• Dipole potential decays as $1/r^2$, faster than the monopole potential ($1/r$)
• Depends on the orientation of the dipole moment relative to the observation point

Quadrupole moment

Quadrupole tensor

• Quadrupole moment is a tensor quantity that describes the distribution of charge in a system with no net charge and no net dipole moment
• Defined as the traceless part of the second moment of the charge distribution: $Q_{ij} = \int \rho(\vec{r}') (3x_i'x_j' - r'^2\delta_{ij}) d^3r'$
• Quadrupole tensor is symmetric and traceless, with nine components

Potential of electric quadrupole

• Electric potential due to a quadrupole at a point $\vec{r}$ is given by: $V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \frac{3}{2} \frac{\hat{r}_i \hat{r}_j Q_{ij}}{r^4}$
• Quadrupole potential decays as $1/r^3$, faster than the dipole potential ($1/r^2$)
• Depends on the orientation of the quadrupole tensor relative to the observation point

Higher order multipole moments

General multipole expansion

• Multipole expansion can be extended to higher-order terms beyond the quadrupole
• Each higher-order term corresponds to a more complex charge distribution
• General expression for the electric potential in terms of multipole moments: $V(\vec{r}) = \frac{1}{4\pi\epsilon_0} \sum_{l=0}^{\infty} \frac{1}{r^{l+1}} \sum_{m=-l}^{l} Q_{lm} Y_{lm}(\theta, \phi)$
• $Q_{lm}$ are the multipole moments, and $Y_{lm}(\theta, \phi)$ are the spherical harmonics

Contribution of higher order terms

• Contribution of higher-order terms decreases rapidly with increasing distance from the charge distribution
• In most practical situations, only the first few terms (monopole, dipole, quadrupole) are significant
• Higher-order terms become important when the charge distribution has a complex structure or when high precision is required

Applications of multipole expansion

Far-field approximation

• Multipole expansion is particularly useful for calculating electric fields and potentials in the far-field region
• Far-field region is defined as the region where the distance from the charge distribution is much larger than its size
• In the far-field, the potential and field can be accurately approximated by the first few terms of the multipole expansion

Electrostatic interactions

• Multipole expansion allows for the calculation of electrostatic interactions between charge distributions
• Interaction energy between two charge distributions can be expressed in terms of their multipole moments
• Multipole moments provide a convenient way to characterize the electrostatic properties of molecules and materials

Multipole expansion in spherical coordinates

Spherical harmonics

• Multipole expansion is often expressed in spherical coordinates using spherical harmonics $Y_{lm}(\theta, \phi)$
• Spherical harmonics are a set of orthonormal functions on the surface of a sphere
• Multipole moments in spherical coordinates are given by: $Q_{lm} = \int \rho(\vec{r}') r'^l Y_{lm}^*(\theta', \phi') d^3r'$

Multipole moments in spherical coordinates

• Monopole moment ($l=0$) is the total charge $Q$
• Dipole moment ($l=1$) has three components corresponding to the Cartesian components of the dipole vector
• Quadrupole moment ($l=2$) has five independent components in spherical coordinates
• Higher-order moments ($l>2$) have $(2l+1)$ components in spherical coordinates

Multipole expansion of magnetic fields

Magnetic scalar potential

• Multipole expansion can also be applied to magnetic fields
• Magnetic scalar potential $\phi_m(\vec{r})$ is analogous to the electric potential
• Expansion of the magnetic scalar potential in terms of magnetic multipole moments: $\phi_m(\vec{r}) = \frac{\mu_0}{4\pi} \sum_{l=0}^{\infty} \frac{1}{r^{l+1}} \sum_{m=-l}^{l} M_{lm} Y_{lm}(\theta, \phi)$

Magnetic multipole moments

• Magnetic monopole moment ($l=0$) is always zero due to the absence of magnetic monopoles
• Magnetic dipole moment ($l=1$) is the first non-vanishing term in the magnetic multipole expansion
• Magnetic quadrupole moment ($l=2$) and higher-order moments can be defined similarly to their electric counterparts

Limitations of multipole expansion

Convergence radius

• Multipole expansion converges only outside a sphere that encloses the entire charge distribution
• Convergence radius is determined by the size of the charge distribution
• Inside the convergence radius, the multipole expansion may not provide accurate results

Accuracy near charge distribution

• Multipole expansion is less accurate when the observation point is close to the charge distribution
• Near the charge distribution, higher-order terms become more significant, and the expansion may require many terms to achieve desired accuracy
• In such cases, alternative methods (e.g., direct integration) may be more appropriate for calculating the potential and field