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Does a Curved Mirror Honestly Reflect Your Identity? A Study of Multipole Images in Front of a Grounded Sphere

Farhang Loran, Saman Moghimi-Araghi

TL;DR

The paper investigates how multipoles image near a grounded conducting sphere behave under the classical image-charge method. By employing Kelvin inversion and Green's-function analysis, it shows that, unlike a flat plane, a multipole’s image generally includes a monopole and all lower-order multipoles, with precise relations depending on orientation and distance. It provides explicit formulas for dipole and quadrupole images, illustrates them with concrete examples, and develops a general framework showing that higher-order multipoles induce lower-order image components in a systematic way. The results deepen the understanding of electrostatic boundary-value problems, offer pedagogical insights into multipole–boundary interactions, and quantify how image charges scale with geometry, with potential implications for controlled field shaping near spherical conductors.

Abstract

The method of image charges is a powerful and elegant technique in electrostatics, commonly used to determine the electric field generated by point charges near conductors of various shapes. While standard problems focus on single charges interacting with conductors, the behavior of multipoles in such configurations has received comparatively less attention, particularly beyond the well-studied case of a flat plane. In this paper, we explore the image formation of electric dipoles and quadrupoles near a conducting sphere and uncover a wonderful result: the image of a given multipole is not necessarily of the same type. Instead, alongside the expected multipole image, the resulting image configuration also includes lower-order multipole contributions. This finding broadens the understanding of electrostatic images and offers new insights into the interaction of multipoles with conducting boundaries.

Does a Curved Mirror Honestly Reflect Your Identity? A Study of Multipole Images in Front of a Grounded Sphere

TL;DR

The paper investigates how multipoles image near a grounded conducting sphere behave under the classical image-charge method. By employing Kelvin inversion and Green's-function analysis, it shows that, unlike a flat plane, a multipole’s image generally includes a monopole and all lower-order multipoles, with precise relations depending on orientation and distance. It provides explicit formulas for dipole and quadrupole images, illustrates them with concrete examples, and develops a general framework showing that higher-order multipoles induce lower-order image components in a systematic way. The results deepen the understanding of electrostatic boundary-value problems, offer pedagogical insights into multipole–boundary interactions, and quantify how image charges scale with geometry, with potential implications for controlled field shaping near spherical conductors.

Abstract

The method of image charges is a powerful and elegant technique in electrostatics, commonly used to determine the electric field generated by point charges near conductors of various shapes. While standard problems focus on single charges interacting with conductors, the behavior of multipoles in such configurations has received comparatively less attention, particularly beyond the well-studied case of a flat plane. In this paper, we explore the image formation of electric dipoles and quadrupoles near a conducting sphere and uncover a wonderful result: the image of a given multipole is not necessarily of the same type. Instead, alongside the expected multipole image, the resulting image configuration also includes lower-order multipole contributions. This finding broadens the understanding of electrostatic images and offers new insights into the interaction of multipoles with conducting boundaries.

Paper Structure

This paper contains 20 sections, 85 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Image Charge Method for a Grounded Conducting Sphere
  3. Image of a dipole in front of a grounded sphere
  4. Case 1: Dipole Perpendicular to the $z$-axis
  5. Case 2: Dipole Parallel to the $z$-axis
  6. The General Case: An Arbitrary Dipole Orientation
  7. Image of Quadrupoles in a Grounded Conducting Sphere
  8. Specific Examples
  9. Example 1.
  10. Example 2.
  11. Example 3.
  12. Potential Due to a Quadrupole
  13. Higher-Order Multipoles
  14. Image of a dipole Parallel to the $z$-axis
  15. Dipole-Induced Potential Near a Grounded Sphere
  16. ...and 5 more sections

Figures (5)

  • Figure 1: The image of a single point charge in the vicinity of a grounded conduction sphere.
  • Figure 2: The configuration of the dipole and the grounded conducting sphere in the first case.
  • Figure 3: The configuration of the dipole and the grounded conducting sphere in the second case.
  • Figure 4: An arbitrary dipole and its image dipole sketched together. Apart from the overall scaling factor, the $z$-component is preserved, while the $x$-component is mirrored with respect to the $z$–axis.
  • Figure 5: Charge configurations corresponding to Examples 1, 2, and 3, illustrated in panels (a), (b), and (c).