Basic Properties of Electric Charges & Coulomb's Law

(i) Additivity of Charge

The total charge of a system is the algebraic sum of all individual charges. Charges add like scalars (with sign):

$Q_{total}=q_1+q_2+q_3+\cdots +q_n$

Example: A system has charges $+3\mu C$, $-5\mu C$, $+2\mu C$.
Total charge $=3-5+2=0\mu C$ (net charge is zero).

(ii) Conservation of Charge

The total electric charge of an isolated system remains constant — charge can neither be created nor destroyed. It can only be transferred from one body to another.

• When a glass rod is rubbed with silk, the glass becomes positive and the silk becomes equally negative — the net charge of the system (glass + silk) remains zero.
• In pair production: $\gamma \to e^{-}+e^{+}$ — a photon creates an electron and a positron, conserving charge (0 → −e + e = 0).
• In pair annihilation: $e^{-}+e^{+}\to 2\gamma$ — charges cancel, producing uncharged photons.
• Conservation of charge holds in all physical processes — nuclear, chemical, and mechanical.

(iii) Quantisation of Charge

Electric charge exists only in discrete multiples of the elementary charge $e$:

$q=ne$     where $n=0,\pm 1,\pm 2,\pm 3,\dots$

The elementary charge: $e=1.6\times {10}^{-19}$ C (charge of one proton or magnitude of charge of one electron).

• Charge of electron: $-e=-1.6\times {10}^{-19}$ C
• Charge of proton: $+e=+1.6\times {10}^{-19}$ C
• Charge of neutron: $0$
• Fractional charges ($e/3$, $2e/3$) exist on quarks but quarks are never found in isolation — only integer multiples of $e$ are observed in free particles.

At macroscopic scales, quantisation is not noticeable because the number of charges is enormous. For example, $1\mu C={10}^{-6}$ C involves ${\displaystyle \frac{{10}^{-6}}{1.6\times {10}^{-19}}}\approx 6.25\times {10}^{12}$ elementary charges.

Summary Table — Three Fundamental Properties

Coulomb's Law gives the electrostatic force between two point charges at rest. Established by Charles-Augustin de Coulomb (1785) through torsion balance experiments:

"The 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. The force acts along the line joining the two charges."

Mathematical Form

$F=k{\displaystyle \frac{|q_1||q_2|}{r^2}}$     where     $k={\displaystyle \frac{1}{4\pi {\epsilon }_0}}=9\times {10}^9$ N·m²·C⁻²

Here ${\epsilon }_0=8.854\times {10}^{-12}$ C²·N⁻¹·m⁻² is the permittivity of free space.

Vector Form of Coulomb's Law

The force on charge $q_2$ due to charge $q_1$ is:

${\overrightarrow{F}}_{12}=k{\displaystyle \frac{q_1{q}_2}{r^2}}{\hat{r}}_{12}$

where ${\hat{r}}_{12}$ is the unit vector pointing from $q_1$ to $q_2$. The sign of $q_1{q}_2$ automatically gives the correct direction:

• $q_1{q}_2>0$ (like charges): force is along $+{\hat{r}}_{12}$ → repulsive.
• $q_1{q}_2<0$ (unlike charges): force is along $-{\hat{r}}_{12}$ → attractive.
• Newton's Third Law holds: ${\overrightarrow{F}}_{12}=-{\overrightarrow{F}}_{21}$ — the forces are equal and opposite.

In a Medium

When charges are in a medium with relative permittivity (dielectric constant) ${\epsilon }_r$ (also written $K$):

$F_{medium}={\displaystyle \frac{1}{4\pi {\epsilon }_0{\epsilon }_r}}{\displaystyle \frac{q_1{q}_2}{r^2}}={\displaystyle \frac{F_{vacuum}}{{\epsilon }_r}}$

Since ${\epsilon }_r\ge 1$ for all media, the force in a medium is always less than or equal to the force in vacuum. For water, ${\epsilon }_r\approx 80$ — electrostatic forces are drastically reduced.

Key Characteristics of Coulomb's Force

When multiple charges are present, the total force on any one charge is the vector sum of the individual forces due to each other charge, calculated using Coulomb's Law independently. The presence of other charges does not affect the force between any two charges.

${\overrightarrow{F}}_{total}={\overrightarrow{F}}_{12}+{\overrightarrow{F}}_{13}+{\overrightarrow{F}}_{14}+\cdots$ $={\displaystyle \sum _{j\ne 1}{\overrightarrow{F}}_{1j}}$

Worked Example — Three Charges in a Line

Three charges $q_1=+4\mu C$, $q_2=-2\mu C$, $q_3=+3\mu C$ are placed along a line. $q_1$ is at origin, $q_2$ at $x=0.3$ m, $q_3$ at $x=0.6$ m. Find net force on $q_2$.

Force on $q_2$ due to $q_1$:
$F_{21}=k{\displaystyle \frac{|q_1||q_2|}{r_{12}^2}}=9\times {10}^9\times {\displaystyle \frac{4\times {10}^{-6}\times 2\times {10}^{-6}}{(0.3{)}^2}}=0.8$ N
Direction: $q_1$ is $+$ve, $q_2$ is $-$ve → attractive → force on $q_2$ is towards $q_1$ → $-\hat{x}$ direction.

Force on $q_2$ due to $q_3$:
$F_{23}=k{\displaystyle \frac{|q_2||q_3|}{r_{23}^2}}=9\times {10}^9\times {\displaystyle \frac{2\times {10}^{-6}\times 3\times {10}^{-6}}{(0.3{)}^2}}=0.6$ N
Direction: $q_3$ is $+$ve, $q_2$ is $-$ve → attractive → force on $q_2$ is towards $q_3$ → $+\hat{x}$ direction.

Net force on $q_2$:
$F_{net}=F_{23}-F_{21}=0.6-0.8=-0.2$ N
Net force = $0.2$ N in $-\hat{x}$ direction (towards $q_1$).

Equilibrium of Charges

For a charge to be in equilibrium, the net force on it must be zero. For a charge placed between two like charges on a line, equilibrium occurs at a point where the two repulsive forces balance. Key results:

• For charges $+Q$ at $x=0$ and $+Q$ at $x=d$: a test charge $+q$ is in equilibrium at $x=d/2$ (midpoint). This is unstable equilibrium.
• For charges $+Q_1$ at origin and $+Q_2$ at $x=d$: the neutral point is at distance $x={\displaystyle \frac{d\sqrt{Q_1}}{\sqrt{Q_1}+\sqrt{Q_2}}}$ from $Q_1$.
• A negative charge placed between two like positive charges can be in equilibrium — and for the system of all three to be in equilibrium, specific charge ratios are required.