Electric Charge and Coulomb's Law

Electric charge is simply the property of matter that makes things push or pull on each other: like charges push apart, unlike ones pull together. Coulomb's law, named after Charles-Augustin de Coulomb, who measured it, says how hard, and it's the starting point for every electric force in this course. All it's really saying is that the push gets weaker the farther apart the charges are. The formula $F = k|q_1q_2|/r^2$ looks technical, but it's just a few simple parts: $q_1q_2$ (how much charge each one has, multiplied together), $r^2$ (the distance between them, squared, so distance hurts a lot) and $k$ (a fixed number that sets the units). Bigger charges, bigger force; more distance, much smaller force. That's the whole law, so bear with the vectors below; they only keep track of which way the push points.

Charge. Rub a balloon on your hair and it clings to the wall; peel tape off a roll and it leaps toward your hand. Rubbing moves electrons from one surface to the other, leaving one with extra electrons (negative) and the other short of them (positive). Like charges repel and unlike charges attract.

• Quantized: every charge is a whole multiple of the elementary charge $e = 1.60\times10^{-19}$ C. A proton carries $+e$ and an electron $-e$. One coulomb is about $6.2\times10^{18}$ electrons' worth, so everyday static charges are measured in µC and nC.
• Conserved: charge is never created or destroyed, only moved. Touch two identical metal spheres together and they share their total charge equally; the total doesn't change.

Coulomb's law. So how hard do two charges push on each other? For small charged objects (point charges), the push or pull runs along the line joining them, with force $F = k|q_1q_2|/r^2$. Double either charge and the force doubles, which makes sense: twice the charge is like two copies of it, each pushing. Pull them apart, though, and the force fades fast. It falls off as the inverse square of the distance, so double $r$ and $F$ drops to a quarter.

Why an inverse square? Picture a charge's influence streaming out evenly in all directions. At distance $r$ it's spread over a sphere of area $4\pi r^2$, so its strength per unit area falls like $1/(4\pi r^2)$. Writing $k = \frac{1}{4\pi\varepsilon_0}$ keeps that sphere in view, and it's why Gauss's Law can say so simply that the field's flux out of any closed surface is $Q/\varepsilon_0$. The exponent itself is an experimental fact: Coulomb measured it with a torsion balance in 1785, and modern tests confirm the 2 to within a few parts in $10^{16}$.

Vector form. With positions $\mathbf{r}_1$ and $\mathbf{r}_2$, the unit vector from charge 1 to charge 2 is $(\mathbf{r}_2 - \mathbf{r}_1)/|\mathbf{r}_2 - \mathbf{r}_1|$. Multiply it by the magnitude $kq_1q_2/|\mathbf{r}_2 - \mathbf{r}_1|^2$ and the powers combine into a cube:

The nice thing about this form is that if you keep the charges' signs in it, it handles direction by itself: $q_1q_2 > 0$ pushes charge 2 away from charge 1, and $q_1q_2 < 0$ pulls it back. Swap the labels and the sign flips, which is just Newton's third law.

Superposition. What if there are more than two charges? Each pair interacts as if the others weren't there, and the forces add as vectors. So for each source charge, find the displacement to the target, work out that force's components, then add the components. It's worth looking for symmetry first, since matching charges placed symmetrically often cancel whole components before you compute anything.

Electric versus gravity. Coulomb's law looks just like Newton's gravity, $Gm_1m_2/r^2$, but between a proton and an electron the electric force is about $2\times10^{39}$ times stronger (Example 2). So why does gravity run the solar system? Because matter is almost perfectly neutral. Its positive and negative charges cancel, and gravity, which only ever attracts, is what's left over at large scales.

A common mistake. Between two unlike charges, it's tempting to look for the spot where the forces balance, since that's where it would be for like charges. It isn't there: between a $+$ and a $-$ charge, a test charge is pushed away from one and pulled toward the other, so both forces point the same way. The balance point is outside the pair, beyond the smaller charge, where the weaker charge is closer and can make up for being weaker. Two smaller slips are worth watching for too. Convert cm to m and µC to $10^{-6}$ C before using $k$, and never add the magnitudes of forces that point different ways. Add components.

Laser printers and photocopiers run on Coulomb attraction: light writes the page as a pattern of charge on a drum, and charged toner powder sticks only where that pattern pulls it, before it's pressed onto the paper. Electrostatic precipitators charge the soot in power-plant and factory exhaust and pull it onto collector plates, catching more than 99% of the ash. Powder-coating lines spray charged paint that wraps around and clings to grounded metal parts, and simulations of drugs binding to proteins add up the Coulomb forces between thousands of atoms.