❓What is the relationship between the force on the charge and the total force on the field for static

What is the relationship between the force on the charge and the total force on the field for static equilibrium?

The relationship between the force on the charge ($\vec{F}_{\text{charge}}$ or $\vec{F}_{\text{Coulomb}}$) and the total force on the field ($\vec{F}_{\text{field}}$ or $\vec{F}_{\text{surface}}$) for a static charge configuration is that they are equal in magnitude and direction.

$\vec{F}_{\text{surface}} = \vec{F}_{\text{Coulomb}}$

In this verification, the surface force on the field ($\vec{F}_{\text{surface}}$) is calculated for the region $x_3 < 0$, which contains the lower charge.

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Verification and Rationale

This relationship is derived from two fundamental principles that must hold for static equilibrium:

1. Static Equilibrium of the Field ($\sum \vec{F}_{\text{field}} = 0$):

   The total force on the electromagnetic field in the lower region ($x_3 < 0$) must be zero. This total force is the sum of the force the charge exerts on the field ($\vec{F}{q \to \text{field}}$) and the force the upper field exerts on the lower field across the surface, which is the computed surface force ($\vec{F}{\text{surface}}$).

   $\vec{F}{q \to \text{field}} + \vec{F}{\text{surface}} = 0 \implies \vec{F}{\text{surface}} = - \vec{F}{q \to \text{field}}$

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Proof and Derivation related to FAQ

What is the relationship between the force on the charge and the total force on the field for static equilibrium? | FAQsviadean on Notion