2.3. In vector form

In vector form:

$\vec{F} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{r}_{12}$

This equation expresses Coulomb's Law in vector form, which incorporates both the magnitude and direction of the force between two point charges. The equation states that the electrostatic force ($\vec{F}$) is proportional to the product of the magnitudes of the charges ($q_1$ and $q_2$) and inversely proportional to the square of the distance between them ($r^2$). The unit vector $\hat{r}_{12}$ indicates the direction of the force, pointing from one charge to the other.

The constant $\epsilon_0$, known as the permittivity of free space, is approximately $8.85 \times 10^{-12} \ C^2/(N \cdot m^2)$ and is used to express the force in terms of SI units.

Think about two magnets: if you hold two magnets in your hands, when you push them together, they either attract or repel based on their poles. The force you feel is analogous to Coulomb’s Law. Just like how the strength and direction of the magnetic force depend on how close the magnets are and how strong they are, the electrostatic force depends on the amount of charge and the distance between charges.

Where: • $\vec{F}$: Electrostatic force • $q_1, q_2$: Point charges • $r$: Distance between the charges • $\epsilon_0$: Permittivity of free space

In this part, each variable in the equation is defined:

• $\vec{F}$ represents the electrostatic force acting on the charges, which is a vector quantity because it has both a magnitude and direction.
• $q_1$ and $q_2$ are the magnitudes of the two point charges involved. They indicate how strong the electric charges are.
• $r$ is the distance between the centers of the two charges. This distance affects how strong the electrostatic force will be; the further apart the charges are, the weaker the force.
• Finally, $\epsilon_0$ is a constant that helps define how electric fields behave in a vacuum. It acts as a proportionality constant in the calculations, ensuring that the units are consistent.

To visualize this, imagine you are lifting two balls, one heavier (larger charge) and one lighter (smaller charge). The effort of lifting depends not just on how heavy the balls are (the charge) but also on how far apart you hold them (the distance). The space around you behaves like a medium (vacuum), which measures how hard it is to lift due to gravity (the permittivity).

The unit vector $\hat{r}_{12}$ indicates the direction of the force, pointing from one charge to the other.

In vector form equations, unit vectors play a key role in conveying direction. The term $\hat{r}_{12}$ is a unit vector that points from charge $q_1$ to charge $q_2$. This means that if $q_1$ and $q_2$ are both positive, they will repel each other, resulting in a force vector that points away from $q_1$ towards $q_2$. Conversely, if one charge is negative, the force would point towards the negative charge since opposite charges attract. Thus, the sign and the direction indicated by the unit vector are crucial for understanding how the charges influence each other.

Consider pushing someone away (like charges repelling) or pulling someone closer (charges attracting). The direction you use to push away or pull in represents the direction of the force. Similarly, the unit vector informs how the charges interact in space.

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