Electrostatics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Electrostatics — Key Facts

Electrostatics deals with stationary electric charges and their effects. All matter contains positive (protons) and negative (electrons) charges. Like charges repel, unlike charges attract.

Coulomb’s Law: $$F = k\frac{q_1 q_2}{r^2} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}$$

where $k = 8.99 \times 10^9$ N·m²/C² and $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m

Force is repulsive if q₁q₂ > 0, attractive if q₁q₂ < 0.

Electric Field: $$E = \frac{F}{q} = k\frac{Q}{r^2}$$

Direction: away from positive, toward negative charge.

⚡ JEE Exam Tip: Electric field is a vector. When calculating total field from multiple charges, use vector addition (component-wise). Electric potential is a scalar — just add algebraically.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Electric Potential:

$$V = k\frac{Q}{r}$$

Relationship: $E = -\frac{dV}{dr}$ or in vector form $\vec{E} = -\nabla V$

For uniform field: $E = V/d$

Superposition Principle: $$\vec{E}_{total} = \sum \vec{E}i, \quad V{total} = \sum V_i$$

Electric Dipole:

A dipole consists of charges +q and -q separated by distance d.

• Dipole moment: $\vec{p} = q\vec{d}$ (direction from -q to +q)
• Torque: $\tau = pE\sin\theta$
• Potential at axial point: $V = \frac{kp}{r^2}$
• Potential at equatorial point: $V = -\frac{kp}{2r^3}$
• Field at axial point: $E = \frac{2kp}{r^3}$

Equipotential Surfaces:

Surfaces where electric potential is constant everywhere.

Properties:

• Electric field is always perpendicular to equipotential surfaces
• No work is done moving a charge along an equipotential surface
• Closer equipotential lines mean stronger electric field

Gauss’s Law:

$$\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\varepsilon_0}$$

This is always true, but particularly useful for symmetric charge distributions.

⚡ JEE Exam Tip: Use Gauss’s law for:

1. Point charge: E = kq/r²
2. Spherical shell: E_inside = 0, E_outside = kq/r²
3. Infinite plane: E = σ/(2ε₀)
4. Uniformly charged sphere: E = (1/4πε₀)(Qr/R³) for r < R

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Applications of Gauss’s Law:

Spherical Shell:

• Inside (r < R): E = 0
• On surface (r = R): E = kQ/R²
• Outside (r > R): E = kQ/r²

The field outside is as if all charge were concentrated at the centre. Inside, the field is zero because enclosed charge = 0.

Infinite Charged Plane Sheet: $$E = \frac{\sigma}{2\varepsilon_0}$$

Direction: perpendicular to plane, away if σ > 0. Note: This field is independent of distance from the plane.

Uniformly Charged Solid Sphere: $$E_{inside} = \frac{1}{4\pi\varepsilon_0}\frac{Qr}{R^3}$$ $$E_{outside} = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}$$

Potential inside: $V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{2R}\left(3 - \frac{r^2}{R^2}\right)$ Potential outside: $V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}$

Electric Dipole in External Field:

When a dipole is placed in a uniform external electric field E:

• Net force = 0 (forces on +q and -q cancel)
• Net torque: τ = pE sinθ
• Potential energy: U = -pE cosθ (minimum when aligned with field)

In non-uniform field, there can be a net force on the dipole.

Energy in Electric Field:

Energy density: $u = \frac{1}{2}\varepsilon_0 E^2$

Energy stored in capacitor: $$U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$$

Capacitance:

Parallel plate: $C = \frac{\varepsilon_0 A}{d}$

With dielectric: $C’ = KC$

Spherical capacitor (inner a, outer b): $C = \frac{4\pi\varepsilon_0 ab}{b-a}$

Dielectric Breakdown:

When electric field exceeds critical value, material conducts:

Material	Breakdown Field (V/m)
Air	3 × 10⁶
Mica	100 × 10⁶
Paper	16 × 10⁶

⚡ JEE Advanced 2023 Analysis: Questions on electric potential due to various charge distributions, energy in capacitors, and dielectrics appeared in recent papers. For conductors in electrostatic equilibrium, the electric field inside is zero and all excess charge resides on the outer surface.

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