Electric Charges and Fields Study Notes

Definition: Electrostatics is the study of forces, fields, and potentials arising from static charges (charges that do not change or move with time).
Historical Context: Thales of Miletus (Greece, around 600 BC) discovered that amber (Greek: elektron) rubbed with wool/silk attracts light objects.
Modern Discovery: Benjamin Franklin named the two types of charges positive and negative. By convention, a glass rod or cat’s fur rubbed with silk/plastic becomes positively charged, while the plastic rod or silk becomes negative.

Electrification: Objects acquire charge via electron transfer; losing electrons makes a body positive, gaining them makes it negative.
Polarity of Charge: The property differentiating the two types of charges.
Basic Law: Like charges repel; unlike charges attract.
Additivity: Total charge of a system is the algebraic sum of all individual point charges.
Conservation: Total charge of an isolated system remains constant; charges are redistributed via transfer but not created or destroyed.
Quantisation: All free charges are integral multiples of a basic unit $e$ . $q = ne$
Constants: The basic unit of charge is $e = 1.602192 \times 10^{-19}\,C$ . In a charge of $-1\,C$ , there are approximately $6 \times 10^{18}$ electrons.

Conductors: Materials that allow electricity to flow easily due to free charges (e.g., metals, humans, Earth).
Insulators: Materials with high resistance that do not allow charge flow (e.g., glass, plastic, wood).
Semiconductors: Materials with resistance intermediate between conductors and insulators.
Grounding: Charges on a conductor flow to the earth when in contact.

Coulomb’s Law: Quantitative force between two point charges ( $q_1, q_2$ ) separated by distance $r$ . $F = k \frac{q_1 q_2}{r^2}$
Constants: In SI units, $k = \frac{1}{4\pi\epsilon_0} \approx 9 \times 10^9\,N\,m^2\,C^{-2}$ . The permittivity of free space is $\epsilon_0 = 8.854 \times 10^{-12}\,C^2\,N^{-1}\,m^{-2}$ .
Principle of Superposition: The net force on a charge is the vector sum of individual forces exerted by other charges, each calculated as if the others were not present.

Electric Field ( $E$ ): Defined as the force per unit test charge $q$ at a point. $E(r) = \frac{1}{4\pi\epsilon_0} \frac{Q}{r^2} \mathbf{\hat{r}}$
Vector Interpretation: Field points radially outward from positive charges and radially inward toward negative charges.
Field Lines: Pictorial mapping where the tangent at any point gives the field direction. Closeness of lines indicates field strength.
Field Line Properties:
- Start at positive charges, end at negative charges.
- Do not cross each other.
- Do not form closed loops.

Electric Flux ( $\Phi$ ): A measure of the number of field lines passing through a given surface area $\Delta S$ . $\Delta\Phi = \mathbf{E} \cdot \Delta\mathbf{S} = E \Delta S \cos(\theta)$
Gauss’s Law: Total electric flux through any closed surface is $1/\epsilon_0$ times the total charge enclosed by the surface. $\Phi = \frac{q_{enclosed}}{\epsilon_0}$

Definition: A pair of equal and opposite charges ( $q, -q$ ) separated by distance $2a$ .
Dipole Moment ( $p$ ): A vector directed from $-q$ to $q$ with magnitude: $p = q \times 2a$
Dipole Field: Falls off as $1/r^3$ at large distances.
Torque ( $\tau$ ): Experienced by a dipole in a uniform external field $\mathbf{E}$ . $\mathbf{\tau} = \mathbf{p} \times \mathbf{E}$

Infinitely Long Wire: Field at distance $r$ with linear charge density $\lambda$ . $E = \frac{\lambda}{2\pi\epsilon_0 r}$
Infinite Plane Sheet: Field with surface charge density $\sigma$ . $E = \frac{\sigma}{2\epsilon_0}$
Thin Spherical Shell:
- Outside ( $r \geq R$ ): $E = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2}$
- Inside (r < R): $E = 0$

Millikan (1912): Experimentally demonstrated the quantisation of charge.
Faraday: Suggested quantisation via laws of electrolysis and introduced the concept of field lines.
Scale Matters: Quantisation is significant at the microscopic level but appears continuous at the macroscopic level (e.g., $1\,\mu C$ contains $10^{13}$ electronic charges).