Coulomb's Law

Coulomb’s Law is one of essential formulae to identify forces occurring in a system of static charges.

Definition §

It is used to determine the magnitude of the Electrostatic Force due to attraction/repulsion between two point charges. A point charge is essentially a point (no length/area/volume) with an electric charge. These differ from Charge Distribution, which are more complex shapes.

Assume we have two point charges, labelled $A$ and $B$.

Coulomb's Law

\vec{F}{q{1} \ on \ q_{2}} = k_e\times\frac{|q_{1}||q_2|}{r^2} \hat{u}= \frac{1}{4\pi\epsilon_0}\times\frac{|q_1||q_2|}{r^2}\hat{u}

>[!terms]- >* $F$ = Electrostatic force ($N$) >* $k_{e}, \frac{1}{4\pi \epsilon_{0}}$ = Coulomb's constant $\approx 8.99\times 10^9$ ($Nm^2C^{-2}$) >* $q_{A},q_{B}$ = Charge at points $A$ and $B$ (in Coulombs, $C$) >* $r$ = Distance between charges ($m$) >* $\epsilon_{0}$ = Permittivity in free space #TOSEE >* $\hat{u}$ = Unit vector representing the direction from $q_{1}$ to $q_{2}$

Explanation §

Essentially, we can now use the sign of the charges as well, to obtain the direction of force. The same principle rule of charges apply - opposites attract and like charges repel:

• If $q_A$ is negative and $q_B$ is negative, $F$ is parallel to $\hat{u}$, showing the force is pushing away from the charges.
• However, if $q_A$ is positive and $q_B$ negative (or vice versa), $F$ is now negative, i.e. is anti-parallel to $\hat{u}$ , showing the charges are attracting

Superposition of Point Charges §

Essentially, if we have multiple point charges in a system, the total electrostatic force on a given charge is the sum of the forces from each individual charge. Pretty obvious. But this superposition is essential when using symmetry to derive forces on continuous charge distributions

$${\vec{F}}_D={\vec{F}}_{AonD}+{\vec{F}}_{BonD}+\dots$$