Surface Force for Two Equal Charges

The electromagnetic field locally mediates the repulsive force between two equal charges ( $+ q$ and $+ q$ ) through tension. On the midplane ( $x^3=0$ ), the total electric field is purely radial ( $E _3=0$ ), meaning the field lines run parallel to the surface. This zero normal component dictates that the total surface force, calculated using the Maxwell stress tensor, results from the tension along these field lines, giving $F= -\frac{q^2}{4 \pi \varepsilon_0(2 d)^2} e_3$. This calculated force is attractive (pulling the two field regions together), and it perfectly balances the upward push that the lower charge exerts on the field, thereby satisfying the requirement for static equilibrium and validating the concept that the field, not just the charges, is a mechanical entity subject to forces.

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Field Tension Mediates Repulsion and Ensures Static Equilibrium#audio

Key takeaways

Electric Field Geometry and Symmetry
- Purely Radial Field: On the $x^3=0$ surface (the midplane), the total electric field $E$ is purely radial ( $E \propto e_\rho$ ). The axial components ( $E_3$ ) from the two equal charges cancel out, while the radial components reinforce.
- Zero Normal Component: Crucially, the component of the electric field normal to the surface is zero ( $E _{ 3 } = 0$ ).
Nature of the Surface Force
- Surface Tension: Because $E_3=0$, the surface force element $d F_3$ is determined solely by the negative pressure term in the Maxwell stress tensor:
$$ d F_3=-\frac{1}{2} \varepsilon_0 E^2 d x^1 d x^2 $$

This term represents tension along the electric field lines. Since the field lines run parallel to the surface, they act like stretched rubber bands, pulling the field regions toward each other.
- Attractive Force: The resulting total surface force $F$ is:
$$ F=-\frac{q^2}{4 \pi \varepsilon_0(2 d)^2} e_3 $$

The negative sign indicates a force pointing downward (in the $-e_3$ direction), meaning the field in the $x^3>0$ region exerts an attractive pull on the field in the $x^3<0$ region.
Consistency with Static Equilibrium and Locality
- Force Balance: The computed surface force $F$ is exactly equal in magnitude and opposite in direction to the force the lower charge exerts on the field. This demonstrates the condition for static equilibrium: the total force on any part of the field must be zero.
- Local Mediation of Repulsion: The problem confirms the principle of locality in electromagnetism. The repulsive force between the two like charges (predicted by Coulomb's law) is physically mediated by the stress within the field:
  - The field is under tension (attractive pull) in the region between the charges.
  - The repulsive push is transmitted through the pressure exerted by the field lines on the charge itself, and the tension exerted by the field lines on the opposite field region.

✍️Mathematical Proof

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