Two point charges, each of charge q, are a distance 2 d...

Two point charges, each of charge $q$, are a distance $2 d$ apart. (a) Find the Maxwell stress tensor $[\mathbf{T}]$ on the plane surface midway between the charges. (b) Find the force on either charge by integrating $[\mathbf{T}] \cdot \mathrm{dS}$ over a plane surface, closing the surface with a large hemisphere of radius $R$. (Show that the integral over the hemisphere vanishes in the limit $R \rightarrow \infty$.) (c) Repeat parts (a) and (b) if the charges have opposite signs. (d) What would the force on either charge be if they were immersed in a simple dielectric of infinite extent?

Two point charges, each of charge $q$, are a distance $2 d$ apart.
(a) Find the Maxwell stress tensor $[\mathbf{T}]$ on the plane surface midway between the charges.
(b) Find the force on either charge by integrating $[\mathbf{T}] \cdot \mathrm{dS}$ over a plane surface, closing the surface with a large hemisphere of radius $R$. (Show that the integral over the hemisphere vanishes in the limit $R \rightarrow \infty$.)
(c) Repeat parts (a) and (b) if the charges have opposite signs.
(d) What would the force on either charge be if they were immersed in a simple dielectric of infinite extent?

Key Concepts

Maxwell Stress Tensor

The Maxwell stress tensor is a second-rank tensor used to describe the flux of electromagnetic momentum through a surface. In electrostatics, it provides a method to compute the forces acting on charges or dielectric bodies by integrating the tensor over a closed surface that encloses the region of interest. It encapsulates the effects of the electric field configuration and is particularly useful when the force cannot be easily obtained by summing pairwise interactions.

Electrostatic Force

Electrostatic force refers to the force between static electric charges, governed by Coulomb's law. This force can be repulsive or attractive depending on the signs of the charges, and its magnitude is inversely proportional to the square of the distance between the charges. In advanced analyses, the force can also be determined by considering the energy and momentum distributions in the electric field via Maxwell’s stress tensor.

Surface Integration for Force Calculation

The method of obtaining forces from the Maxwell stress tensor involves calculating a surface integral over a closed surface that encloses the charge or region of interest. This integration transforms local field stresses into a net force on the contained body. The approach is particularly powerful when direct computation of forces through other means is cumbersome, especially in complex geometries or in the presence of symmetry.

Asymptotic Analysis in Electromagnetism

In many problems involving fields extending to infinity, such as the integration over a large hemisphere or other surfaces, it is necessary to analyze the behavior of the integrals as the radius or extent of the region goes to infinity. Asymptotic analysis is the study of this behavior, ensuring that contributions from such regions vanish or simplify, which is key to ensuring that only the local field contributions determine the net force.

Dielectric Medium Effects

When charges are immersed in a dielectric medium, the electric field and resulting forces are modified by the material's dielectric constant. The presence of a dielectric changes the effective permittivity, thus altering the strength of the electric interactions. This is important when evaluating forces using Maxwell’s stress tensor or Coulomb's law in non-vacuum media, as the material's ability to reduce the electric field strength directly impacts the magnitude of the force experienced by the charges.

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