Electrostatics and Electric Field

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Electrostatics — Key Facts

Electrostatics studies electric charges at rest. Like charges repel, unlike charges attract. The SI unit of charge is Coulomb (C), and the elementary charge is e = 1.6 × 10⁻¹⁹ C.

Coulomb’s Law: The force between two point charges: $$F = k\frac{q_1 q_2}{r^2} = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r^2}$$

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: The electric field strength at a point is the force per unit charge: $$E = \frac{F}{q} = k\frac{Q}{r^2}$$

Direction: away from positive charge (Q > 0), toward negative charge (Q < 0)

Electric Potential: $$V = k\frac{Q}{r}$$

Relationship between E and V: $E = -\frac{dV}{dr}$

For uniform field: $E = V/d$

⚡ ECAT Exam Tip: Electric potential is a scalar (can be positive or negative), while electric field is a vector. Always use vector addition for E fields but algebraic addition for V.

---

🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Superposition Principle:

For multiple charges, total field/potential is vector/scalar sum: $$\vec{E}_{total} = \sum \vec{E}i$$ $$V{total} = \sum V_i \quad \text{(algebraic sum)}$$

Electric Dipole:

An electric dipole consists of two equal and opposite charges (-q, +q) separated by distance d.

• Dipole moment: $p = qd$ (direction from -q to +q, unit: C·m)
• Torque on dipole in uniform field: $\tau = pE\sin\theta$
• Potential at axial point (along dipole axis, distance r): $V = \frac{kp}{r^2}$
• Potential at equatorial point (perpendicular): $V = -\frac{kp}{2r^3}$
• Field at axial point: $E = \frac{2kp}{r^3}$

Gauss’s Law:

The total electric flux through any closed surface: $$\oint \vec{E} \cdot d\vec{A} = \frac{q_{enc}}{\varepsilon_0}$$

This is equivalent to Coulomb’s law and is especially useful for symmetric charge distributions.

Applications of Gauss’s Law:

Charge Distribution	Electric Field
Point charge	$E = kq/r^2$ (all space)
Spherical shell (radius R)	Inside: $E = 0$; Surface: $E = kq/R^2$; Outside: $E = kq/r^2$
Infinite plane sheet	$E = \frac{\sigma}{2\varepsilon_0}$ (constant, independent of distance)
Uniformly charged sphere	Inside: $E = \frac{kQr}{R^3}$; Outside: $E = kQ/r^2$
Parallel plate capacitor	$E = \frac{\sigma}{\varepsilon_0} = \frac{V}{d}$ (uniform field)

Dielectrics:

When a dielectric (insulating material) is placed in an electric field:

• Polar dielectrics: molecules align with field
• Non-polar dielectrics: induced dipoles
• Effective field: $E_{eff} = E_0/K$ where K is dielectric constant
• Capacitance increases: $C’ = KC$

⚡ ECAT Exam Tip: For ECAT, remember that the field inside a conductor in electrostatic equilibrium is zero. All excess charge resides on the outer surface. This is why Faraday cages work.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Energy in Electric Field:

Energy density in electric field (in vacuum): $$u = \frac{1}{2}\varepsilon_0 E^2$$

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

Capacitance:

Parallel plate capacitor: $C = \frac{\varepsilon_0 A}{d}$ With dielectric: $C = \frac{K\varepsilon_0 A}{d}$

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

Cylindrical capacitor (length L): $C = \frac{2\pi\varepsilon_0 L}{\ln(b/a)}$

Capacitor Combinations:

Series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + …$ Parallel: $C_{eq} = C_1 + C_2 + …$

Dielectric Breakdown:

When electric field exceeds critical value, material becomes conductive:

Material	Dielectric Strength (V/m)
Air	$3 \times 10^6$
Mica	$100 \times 10^6$
Paper	$16 \times 10^6$

Maximum voltage before breakdown: $V_{max} = E_{breakdown} \times d$

Method of Images:

For solving electrostatic problems with conductors:

• Replace induced charges with image charges
• Image of point charge q at distance d from infinite grounded plane: image charge -q at distance d behind the plane

Uniqueness Theorem:

Given the charges within a region and the potential on the boundaries, the electric field is uniquely determined. This justifies the method of images.

⚡ ECAT 2024 Analysis: Questions on electric field due to spherical shells, parallel plate capacitors, and Gauss’s law applications appear frequently. The relationship E = V/d for uniform fields and the fact that E inside a conductor is zero are key concepts.

---

Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.