❓What is the Coulomb force exerted by one charge on the other for two equal charges separated by 2d?

The Coulomb force ($\vec{F}_{\text{Coulomb}}$) exerted by one charge ($q$) on the other ($q$) is a repulsive force.

If we consider the force exerted on the lower charge ($q$ at $x_3 = -d$) by the upper charge ($q$ at $x_3 = d$), the vector force is: $\vec{F}_{\text{Coulomb}} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{r}$, With $q_1 = q_2 = q$, distance $r = 2d$, and the unit vector pointing from the source (upper charge) to the test charge (lower charge) being $-\vec{e}3$ (downward): $\vec{F}{\text{Coulomb}} = -\frac{q^2}{4\pi\epsilon_0 (2d)^2} \vec{e}_3$

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Magnitude and Direction

• Magnitude: $\left|\vec{F}_{\text{Coulomb}}\right| = \frac{q^2}{4\pi\epsilon_0 (2d)^2}$

• Direction: The force is in the $-\vec{e}_3$ direction (downward), which correctly indicates repulsion for two equal, positive charges separated along the $x_3$-axis.

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Proof and Derivation related to FAQ

What is the Coulomb force exerted by one charge on the other for two equal charges separated by 2d? | FAQsviadean on Notion