Coulomb’s Law: MDCAT Physics notes
Coulomb’s Law MDCAT notes: force between point charges, inverse square law, effect of medium and relative permittivity, and ratio problems solved fast.
Statement
The electrostatic force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. It acts along the line joining the charges.
$$F = k\frac{q_1q_2}{r^2}, \qquad k = \frac{1}{4\pi\varepsilon_0} = 9\times10^9\ \mathrm{N\,m^2\,C^{-2}}$$
$\varepsilon_0 = 8.85\times10^{-12}$ C$^2$ N$^{-1}$ m$^{-2}$ is the permittivity of free space.
- Like charges repel, unlike charges attract.
- With signs included, a positive $F$ means repulsion and a negative $F$ means attraction.
- The forces on the two charges are equal and opposite, so Coulomb's law obeys Newton's third law.
The constant $k$
$k$ depends on both the medium between the charges and the system of units. It does not depend on the type or size of the charges.
Effect of a medium
If the space between the charges is filled with an insulator (dielectric) of relative permittivity $\varepsilon_r$:
$$F_{\text{med}} = \frac{1}{4\pi\varepsilon_0\varepsilon_r}\frac{q_1q_2}{r^2} = \frac{F_{\text{vac}}}{\varepsilon_r}$$
The force is reduced by the factor $\varepsilon_r$ (also called the dielectric constant). If the force in a medium becomes one-fifth, $\varepsilon_r = 5$. For air $\varepsilon_r \approx 1$; for water $\varepsilon_r \approx 80$.
Why salt dissolves in water: water's high $\varepsilon_r$ weakens the attraction between Na$^+$ and Cl$^-$ ions to about 1/80 of its value in vacuum, so the ions separate easily.
Ratio problems
Write $F \propto q_1q_2/r^2$ and change one factor at a time.
| Change | New force |
|---|---|
| $r$ doubled | $F/4$ |
| $r$ halved (decreased by 50%) | $4F$ |
| $r$ tripled | $F/9$ |
| Both charges doubled | $4F$ |
| Both charges doubled and $r$ halved | $16F$ |
| One charge doubled, $r$ doubled | $F/2$ |
Example: two charges attract with 12 N. If the separation is doubled, the force is $12/4 = 3$ N. If instead a medium of $\varepsilon_r = 4$ fills the gap, the force is $12/4 = 3$ N as well.
Electric vs gravitational force
| Feature | Electrostatic | Gravitational |
|---|---|---|
| Acts on | Charge | Mass |
| Nature | Attractive or repulsive | Only attractive |
| Law | Inverse square | Inverse square |
| Depends on medium | Yes | No |
| Strength | Very much stronger | Very much weaker |
Charge and induction
Charge is the property of matter that makes it attract or repel other charged objects. A charged body can attract a neutral one by electrostatic induction: it pulls opposite charge to the near side of the neutral body, and the nearer opposite charge is attracted more strongly than the farther like charge is repelled.
Repulsion is the sure test of charge: if A repels B and B repels C, all three carry the same sign. Attraction alone cannot prove a body is charged.
Key formulas
- $F = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1q_2}{r^2}$
- $F_{\text{med}} = F_{\text{vac}}/\varepsilon_r$
- $k = 9\times10^9$ N m$^2$ C$^{-2}$
Common MDCAT traps
- Force depends on the product of charges, not the sum.
- "Decreased by 50%" means $r$ is halved, so the force becomes four times, not double.
- A dielectric decreases the force by $\varepsilon_r$; it never increases it.
- $k$ depends on medium and units; choose the combined option.
- Electric forces do not act on mass; only gravity is purely attractive.
Quick revision
- $F \propto q_1q_2$ and $F \propto 1/r^2$.
- Positive Coulomb force means repulsion.
- Water's high permittivity dissolves ionic compounds.
- Coulomb forces form a Newton's third law pair.
- Gravity is far weaker than the electric force between elementary particles.