Electrostatics – Complete Exam-Oriented Notes

Electric Charge: Nature, Properties & Units

Electric charge ($Q$) is a fundamental, intrinsic property of matter producing electrostatic interactions.

• Types: $+ve$ (deficit of electrons), $-ve$ (excess of electrons), neutral ($\text{no. of }e^- = \text{no. of }p^+$).

• Elementary charge: $e = 1.6\times10^{-19}\,\text{C}$. Minimum observable charge: $\pm e$. Quarks appear to carry $\pm\tfrac{1}{3}e,\,\pm\tfrac{2}{3}e$ but are not free; thus electron’s charge defines quantisation.

• Quantisation: $Q = \pm n e,\;n=0,1,2,3…$ Charges like $0.1e,\,\tfrac{5}{24}e$ are impossible.

• Conservation: In an isolated system $\Sigma Q$ remains constant; creation / annihilation occur in equal and opposite pairs (pair-production etc.).

• Invariance: Electric charge is speed-independent (mass increases relativistically; charge does not).

• Scalar nature: Algebraic (not vector) addition.

• Units
– SI: coulomb (C)
– CGS-esu (stat-coulomb), emu, franklin (Fr), faraday (Fdy).

• Specific charge: $S=\dfrac{Q}{m}$ (speed-dependent only through mass).

Methods of Charging

1. Friction (triboelectric): equal & opposite charges appear on insulators.
2. Conduction: physical contact; same sign appears on originally neutral conductor; final charge divides $\propto$ radii $R$.
3. Induction: nearby charged body polarises a conductor/dielectric. – Near surface acquires opposite sign (equal for conductor, lesser for dielectric). – Gold-leaf electroscope exploits induction.

Electroscope

Gold-leaf electroscope detects (not measures) charge. Divergence $\theta$ depends on net charge delivered. Comparative deflections example: $+20\,\text{C}\to5^\circ, -40\,\text{C}\to9^\circ \Rightarrow +30\,\text{C}\to7^\circ$ (approx. proportional to magnitude).

Coulomb’s Law

For two point / spherically symmetrical charges $q<em>1, q</em>2$ separated by $r$ in a medium of relative permittivity $\varepsilon<em>r$, $F = k\frac{|q</em>1 q<em>2|}{r^2},\; k = \frac{1}{4\pi\varepsilon</em>0\varepsilon<em>r}$. – Vector form: $\vec F</em>{12}=k\frac{q<em>1q</em>2}{r<em>{12}^2}\hat r</em>{12}$ (repulsive if q1q2>0, attractive otherwise).
– Obeys inverse-square, action-reaction, conservative, infinite range, photon mediation.

Superposition: Net force/field is vector sum; force of one charge on another is unaffected by presence of additional charges (but net force is medium-dependent).

Typical Quantitative Results & Checks

• Equal and opposite charges (e.g., $+2\,\mu\text{C},-3\,\mu\text{C}$) experience equal magnitude forces (ratio 1:1).
• Scaling: If each charge $\to\tfrac12$ and distance $\to2r$, $F' = \tfrac{1}{16}F$.
• Charge sharing: Bring spheres of charges $q<em>1,q</em>2$ into contact, separate—common potential makes $q'\propto R$.

Electric Field (E-field)

Definition: $\vec E = \dfrac{\vec F}{q<em>0}$ on small positive test charge $q</em>0$. Units: $\text{N/C}=\text{V/m}$.

Point charge: $E = k\dfrac{Q}{r^2}$ radially outward (Q>0) or inward (Q<0). Field undefined at location of source.

Field of Charge Configurations

• Multiple point charges: vector summation (examples: square corner charges, GP on x-axis).
• Ring of radius R, centre on axis distance x: $E<em>x = kQx/(R^2+x^2)^{3/2}$, $E</em>{centre}=0$; far-field $E\approx kQ/x^2$.
• Circular arc $\theta$ (rad): $E = 2k\lambda\sin(\theta/2)/R$ along bisector. Half-ring $\theta=\pi$ gives $E = 2k\lambda/\pi R$.
• Infinite line charge $\lambda$: $E = 2k\lambda/R$ (radially). Semi-infinite line: $E = k\lambda/R$.
• Large non-conducting sheet (surface density $\sigma$): $E = \sigma/2\varepsilon<em>0$ on either side. • Conducting sheet: $E = \sigma/\varepsilon</em>0$ normal outwards (field zero inside conductor).
• Solid non-conducting sphere (uniform $\rho$): $E<em>{inside}=kQr/R^3,\;E</em>{surface}=kQ/R^2,\;E<em>{outside}=kQ/r^2$. • Conducting sphere: $E</em>{inside}=0,\;E<em>{surface}=kQ/R^2,\;E</em>{outside}=kQ/r^2$.
• Cavity inside uniformly charged solid: uniform field $E=\rho r/3\varepsilon_0$.

Electric Field Lines (Force Lines)

Imaginary curves tangent to $\vec E$ at every point.
Rules: begin on $+$, end on $-$; never intersect; density $\propto |\vec E|$; enter conductor perpendicular; absent inside conductor; never form closed loops; not identical to path of test charge (parabolic vs straight examples).
Comparative questions (density near A vs B etc.) follow these rules.

Neutral / Zero-Field Points

For two like charges $q, n q$ on line, point where $E=0$ lies externally on side of smaller charge at $x = r\sqrt{n}+?$:
Same sign: $x=\dfrac{r}{\sqrt n -1}$ measured from smaller charge. Opposite sign: $x=\dfrac{r}{\sqrt{n}+1}$ (between them). Numerous MCQ solved (25 µC & 36 µC, etc.).

Pendulum-Type Problems

Two identical masses $m$ with charge $q$ suspended by length $l$: equilibrium $\tan\theta = \dfrac{kq^2}{mg(2l\sin\theta)}$; small-angle approximations yield $\theta\propto q$. Inside satellite ($g=0$), tension $T = kq^2/4l^2$ only. When strings clamped half-height (NEET 2013) new separation $=(3/\sqrt 2)^{1/3}r$.

Motion of Charges in Uniform $\vec E$

• Rest release: acceleration $a = qE/m$, velocity $v=\sqrt{2qEy/m}$ after distance y, time $t=\sqrt{2my/qE}$.
• Projection parallel to $E$: straight-line kinematics $v=u\pm a t$.
• Projection perpendicular: parabolic trajectory analogous to projectile under gravity: $y=\tfrac{qE}{2m}x^2/u^2$.
Applications: proton/time in field, electron fall vs proton fall (NEET 2018), toy-car average velocity, proton–alpha curvature comparison etc.

Electric Dipole

Two equal & opposite charges $\pm q$ separated by $2a$; dipole moment $\vec p = q\,2a\,\hat\ell$ (from − to +).
Composite arrangements (triangular, 120°, etc.) resolved vectorially ($p_{\text{net}} = \sqrt3 ql$ etc.).

Electric Field of Dipole

• Axial point distance $r$ (with $r\gg a$): $E<em>{ax} = \dfrac{2kp}{r^3}$ along $\vec p$. • Equatorial point: $E</em>{eq}=\dfrac{kp}{r^3}$ opposite $\vec p$.
• General angle $\theta$: $E = \dfrac{kp}{r^3}\sqrt{1+3\cos^2\theta}$; perpendicular component $E_\perp = kp\sin\theta/r^3$.

Force Interactions

• Point charge on dipole axis: $F \propto 1/r^3$; doubling r reduces $F$ by $1/8$.
• Dipole–dipole: $F,\tau \propto 1/r^4$.

Dipole in Uniform $\vec E$

– Net force zero; torque $\vec\tau = \vec p \times \vec E = pE\sin\theta\,\hat n$.
– Potential energy $U = -pE\cos\theta$; stable equilibrium $\theta=0$ (minimum), unstable $\theta=\pi$ (maximum).
– Small oscillations: $\displaystyle \omega = \sqrt{\dfrac{pE}{I}},\; T = 2\pi\sqrt{\dfrac{I}{pE}}$.

– Work in rotation: $W<em>{\theta</em>1\to\theta<em>2}=pE(\cos\theta</em>1-\cos\theta_2)$. Sample: from $0\to90^\circ$ gives $pE$; $0\to180^\circ$ gives $2pE$ (AIPMT, NEET questions).

– Dipole in non-uniform field: experiences force $\vec F = (\vec p \cdot \nabla)\vec E$ in addition to torque; moves toward lower potential energy (charges experience different magnitudes).

Electric Flux ($\Phi_E$)

Flux through surface $S$: $\Phi<em>E = \displaystyle\iint</em>S \vec E\cdot d\vec A$.

Simple cases: Flat area with uniform $E$: $\Phi = EA\cos\theta$.
Hemispherical surface in uniform $E$: net $\Phi = E \pi R^2$ (curved surface cancels base). Parallel-axis cylinder: $\Phi=0$ (in & out equal). Numerous MCQ applications to half-covered square, tilted rectangles, etc.

Gauss’s Law

$\displaystyle \oint<em>S \vec E \cdot d\vec A = \frac{q</em>{\text{in}}}{\varepsilon_0}$ for any closed (Gaussian) surface.

Key notes:

1. Independent of external charges; only enclosed charge matters; location inside doesn’t affect flux; if $q_{in}=0$, $\Phi=0$.
2. Useful when symmetry (spherical, cylindrical, planar) lets $E$ be constant on Gaussian surface.
3. Gaussian surface must not pass through point charges (field undefined there).
4. Permits quick flux division problems (charge at cube centre: each face $q/6\varepsilon<em>0$; at corner: through a cube $q/8\varepsilon</em>0$ hence one face $q/24\varepsilon_0$, etc.).

Fields via Gauss

• Sphere around point charge: retrieves Coulomb.
• Infinite line ($\lambda$): Cylindrical Gaussian, $E=\lambda/2\pi\varepsilon<em>0 r$. • Infinite plane sheet ($\sigma$): Pill-box gives $E=\sigma/2\varepsilon</em>0$; conducting sheet $E=\sigma/\varepsilon<em>0$ (field only outside). • Solid non-conducting sphere: $E</em>{inside}=\rho r/3\varepsilon_0$.
• Cavity, concentric shells, concentric spheres (charge only on outer shell) etc. handled similarly.

Conductor vs Insulator

Conductors: free electrons move; net $E=0$ inside in electrostatic equilibrium, charge resides on outer surface, denser at sharp points.
Insulators (dielectrics): electrons bound; polarisation occurs but no free conduction.

Sample Numerical Gems

• Electrons removed $N\Rightarrow Q=Ne$ (e.g., $10^{14}$ e removed $\Rightarrow Q=+16\,\mu\text{C}$).
• $1\,\text{C}$ equals $6.25\times10^{18}$ electrons.
• Charge possibilities: multiples of $e$; $1.6\times10^{-20}\,\text{C}$ impossible.
• $\alpha$-particle charge $+2e=3.2\times10^{-19}\,\text{C}$.

Miscellaneous Conceptual Points

• Sure test for charge: repulsion (attraction can occur with induction even if uncharged).
• Mass change on charging: add/remove electrons, mass changes by $\Delta m=N m_e\approx10^{-3}\,\text{mg}$ scale.
• Energy vs charge: cannot convert energy ↔ charge (charge conserved).
• Field inside hollow conductor remains zero regardless of external fields (shielding).
• Field lines around induced neutral conductor in external field distort but net enclosed charge unchanged.

Ethical & Practical Relevance

Electrostatic principles underpin particle accelerators, xerography, ink-jet printing, pollution precipitators, and medical defibrillators; understanding field shielding is vital for spacecraft, electronics enclosures, MRI rooms, etc.

These notes encapsulate every theme, law, derivation, qualitative rule, and standard formula discussed across the full transcript, interleaving worked-example insights and exam-centric emphases (AIPMT/NEET). They serve as a comprehensive standalone reference for Electrostatics: charges, fields, Gauss applications, dipole physics, flux concepts, and motion of charges.