Interaction of Charge Dipoles

Paper: Interaction of a Charge Dipole within the Near Field of a Larger Dipole

Abstract

This paper explores the interaction between a smaller charge dipole and the near field of a larger, lower-frequency dipole. The study focuses on the mathematical description of the electromagnetic fields involved, the forces and torques exerted on the smaller dipole, and the alignment of the dipole planes. The interaction is modeled using fundamental principles of electromagnetism and quantum mechanics.

Introduction

The concept of charge dipoles interacting within the near field of other dipoles is central to understanding various electromagnetic phenomena. The analysis includes the temporal aspects of the charge dipole and its alignment within the electromagnetic environment.

Electromagnetic Fields in the Near Field

The near field of a dipole refers to the region close to the dipole where electromagnetic fields exhibit complex, non-radiative behavior.

Electric Field (E-field)

The near-field electric field of a dipole is given by:

E(r) = (1 / (4*πϵ0)) * ((3*(p⋅r)*r) / (r5 – p / r3)

where p is the dipole moment, r is the position vector relative to the dipole, and r=∣r∣.

Magnetic Field (H-field)

The near-field magnetic field of a dipole is given by:

H(r) = (1 / (4*π) ) * ((3*(m⋅r)*r) / (r5 – m / r3)

where m is the magnetic dipole moment.

Interaction with a Smaller Dipole

The smaller dipole within the near field of the larger dipole experiences forces and torques due to the electromagnetic fields.

Force on the Smaller Dipole

The force F on a dipole ps in an electric field E is given by:

F = (ps ⋅∇) E

Torque on the Smaller Dipole

The torque T on a dipole ps in an electric field E is given by:

T = ps × E

Alignment of Dipole Planes

The alignment of the planes of the dipoles is influenced by the torque exerted on the smaller dipole. The orientation can be described using the potential energy U of the dipole in the field:

U = −ps ⋅ E

Mathematical Description

To describe the full interaction mathematically, we combine the fields and forces.

Electric Field of the Larger Dipole

The electric field of the larger dipole at a position rs​ of the smaller dipole is:

EL(rs) = 1 / 4πϵ0 ( 3 ( pL⋅rs) rs / rs5 −pL / rs3 )

The force on the smaller dipole at position rs​ is:

Fs = ( ps⋅∇ ) EL (rs)

Torque on the Smaller Dipole

The torque on the smaller dipole at position rs​ is:

Ts = ps × EL (rs)

Potential Energy and Alignment

The potential energy of the smaller dipole at position rs​ is:

Us = −ps ⋅ EL (rs)

Influence of the Surrounding Environment

The surrounding environment affects the smaller dipole’s energy and orientation.

Induced Dipole Moment

If the smaller dipole’s moment ps​ is induced by the larger dipole’s field, we have:

ps = α EL (rs)

where αα is the polarizability of the smaller dipole.

Resulting Interaction Energy

The resulting interaction energy is:

Uint = −α ( EL (rs) ⋅ EL (rs) )

Conclusion

The interaction of a smaller charge dipole within the near field of a larger dipole involves calculating the local electric and magnetic fields, determining the forces and torques on the smaller dipole, and understanding how the alignment and potential energy of the dipoles are influenced by their relative positions and orientations. This mathematical framework provides a comprehensive understanding of the complex interactions between dipoles in an electromagnetic environment.

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