❓How is the computed surface force on the field for equal charges verified using Coulomb's Law and th

How is the computed surface force on the field for equal charges verified using Coulomb's Law and the condition for static equilibrium?

The computed surface force on the field for two equal charges is verified using Coulomb's Law and the condition for static equilibrium by ensuring the forces on the lower field region cancel out.

The verification process follows these steps:

1. Calculate the Coulomb Force ($\vec{F}_{\text{Coulomb}}$)

The Coulomb force exerted by the upper charge ($x_3 = d$) on the lower charge ($q$ at $x_3 = -d$) is a repulsive force pointing downward ($-\vec{e}_3$).

$\vec{F}{\text{Coulomb}} = \vec{F}{\text{field} \to q} = -\frac{q^2}{4\pi\epsilon_0 (2d)^2} \vec{e}_3$

This is the force exerted by the field on the lower charge.

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2. Determine the Force of the Charge on the Field ($\vec{F}_{q \to \text{field}}$)

By Newton's Third Law, the force exerted by the lower charge on the field in its vicinity is equal in magnitude and opposite in direction to the force exerted by the field on the charge:

$\vec{F}{q \to \text{field}} = -\vec{F}{\text{field} \to q} = -\vec{F}_{\text{Coulomb}}$

Substituting the result from Step 1:

$\vec{F}_{q \to \text{field}} = - \left( -\frac{q^2}{4\pi\epsilon_0 (2d)^2} \vec{e}_3 \right) = \frac{q^2}{4\pi\epsilon_0 (2d)^2} \vec{e}_3$

This force points upward ($+\vec{e}_3$).

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How is the computed surface force on the field for equal charges verified using Coulomb's Law and the condition for static equilibrium? | FAQsviadean on Notion