Coulomb’s Law

• Force between two point charges:

  \[\mathbf{F} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{r} = k \frac{q_1 q_2}{r^2} \hat{r}\]

• \(k = \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \, \text{N.m^2/C^2}\)

• \(\epsilon_0 = 8.854 \times 10^{-12} \, \text{F/m}\) (permittivity of free space)

• Key Points:

  • Inverse square law

  • Like charges repel, unlike charges attract

  • Superposition principle applies

Electric Field Intensity

• Electric field due to a point charge:

  \[\mathbf{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}\]

• Superposition principle: \(\mathbf{E}_{\text{total}} = \sum \mathbf{E}_i\)

• Units: N/C or V/m

• For continuous charge distributions:

  \[\mathbf{E} = \frac{1}{4\pi\epsilon_0} \int \frac{dq}{r^2} \hat{r}\]

• Direction: Away from positive, towards negative charges

Electric Flux Density

• Defined as: \(\mathbf{D} = \epsilon_0 \mathbf{E}\) (in vacuum)

• In dielectric medium: \(\mathbf{D} = \epsilon \mathbf{E} = \epsilon_0 \epsilon_r \mathbf{E}\)

• Units: C/m²

• Key Properties:

  • Independent of medium properties

  • Continuous across dielectric boundaries

  • Useful in Gauss’s Law applications

• Electric flux: \(\Phi = \int \mathbf{D} \cdot d\mathbf{S}\)

Gauss’s Law

• Integral form: \(\oint \mathbf{D} \cdot d\mathbf{S} = Q_{\text{enc}}\)

• Differential form: \(\nabla \cdot \mathbf{D} = \rho\)

• For vacuum: \(\oint \mathbf{E} \cdot d\mathbf{S} = \frac{Q_{\text{enc}}}{\epsilon_0}\)

• Applications:

  • Spherical symmetry: Choose spherical Gaussian surface

  • Cylindrical symmetry: Choose cylindrical Gaussian surface

  • Planar symmetry: Choose box-shaped Gaussian surface

• GATE Tip: Always exploit symmetry!

Divergence

• Measures field’s divergence from a point:

  \[\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \quad \text{and} \quad \nabla \cdot \mathbf{D} = \rho\]

• In Cartesian coordinates:

  \[\nabla \cdot \mathbf{E} = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z}\]

• Physical interpretation: Source/sink of field lines

• Key Point: Positive divergence = source, Negative = sink

Point Charge Distribution

• Potential: \(V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}\)

• Electric field: \(\mathbf{E} = -\nabla V = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}\)

• For multiple point charges:

  \[V = \frac{1}{4\pi\epsilon_0} \sum_i \frac{q_i}{r_i}\]

• Key Relations:

  • \(\mathbf{E} = -\nabla V\)

  • \(V = -\int \mathbf{E} \cdot d\mathbf{l}\)

  • Work done: \(W = q(V_f - V_i)\)

Line Charge Distribution

• Linear charge density: \(\lambda = \frac{dq}{dl}\)

• For infinite line charge along z-axis:

  \[\mathbf{E} = \frac{\lambda}{2\pi\epsilon_0 \rho} \hat{\rho} \quad \text{(cylindrical coords)}\]

• Potential (reference at \(\rho = \rho_0\)):

  \[V = \frac{\lambda}{2\pi\epsilon_0} \ln\left(\frac{\rho_0}{\rho}\right)\]

• GATE Note: Potential reference point is crucial!

Plane Charge Distribution

• Surface charge density: \(\sigma = \frac{dq}{dS}\)

• For infinite plane sheet:

  \[\mathbf{E} = \frac{\sigma}{2\epsilon_0} \hat{n}\]

• Field is uniform and perpendicular to plane

• Potential: \(V = -\frac{\sigma}{2\epsilon_0} |z|\) (for sheet at z=0)

• Two parallel plates:

  \[\mathbf{E} = \frac{\sigma}{\epsilon_0} \hat{n} \quad \text{(between plates)}\]

Spherical Charge Distribution

• Volume charge density: \(\rho = \frac{dq}{dV}\)

• For uniform sphere of radius \(a\):

• Outside sphere (\(r > a\)):

  \[\mathbf{E} = \frac{Q}{4\pi\epsilon_0 r^2} \hat{r}, \quad V = \frac{Q}{4\pi\epsilon_0 r}\]

• Inside sphere (\(r < a\)):

  \[\mathbf{E} = \frac{Qr}{4\pi\epsilon_0 a^3} \hat{r}, \quad V = \frac{Q}{4\pi\epsilon_0 a}\left(\frac{3}{2} - \frac{r^2}{2a^2}\right)\]

• At surface (\(r = a\)): \(V = \frac{Q}{4\pi\epsilon_0 a}\)

Effect of Dielectric Medium

• Dielectric reduces electric field:

  \[\mathbf{D} = \epsilon \mathbf{E} = \epsilon_0 \epsilon_r \mathbf{E}\]

• \(\epsilon_r\): Relative permittivity (dielectric constant)

• Polarization vector: \(\mathbf{P} = \epsilon_0 (\epsilon_r - 1) \mathbf{E}\)

• Relationship: \(\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}\)

• Boundary Conditions:

  • Normal component: \(D_{1n} - D_{2n} = \sigma_s\)

  • Tangential component: \(E_{1t} - E_{2t} = 0\)

Dielectric Boundary Conditions

• No free surface charge (\(\sigma_s = 0\)):

  • \(D_{1n} = D_{2n}\) (normal D continuous)

  • \(E_{1t} = E_{2t}\) (tangential E continuous)

• Refraction of field lines:

  \[\frac{\tan \theta_1}{\tan \theta_2} = \frac{\epsilon_1}{\epsilon_2}\]

• GATE Application: Capacitor with multiple dielectrics

• Fields in different media related by permittivity ratios

Capacitance of Simple Configurations

• Definition: \(C = \frac{Q}{V}\), Units: F (Farad)

• Parallel plate capacitor:

  \[C = \frac{\epsilon A}{d} = \frac{\epsilon_0 \epsilon_r A}{d}\]

• Cylindrical capacitor: (inner radius \(a\), outer radius \(b\))

  \[C = \frac{2\pi\epsilon l}{\ln(b/a)}\]

• Spherical capacitor: (inner radius \(a\), outer radius \(b\))

  \[C = 4\pi\epsilon \frac{ab}{b-a}\]

Capacitance - Special Cases & Combinations

• Isolated conducting sphere: \(C = 4\pi\epsilon a\)

• Effect of dielectric: \(C_{\text{new}} = \epsilon_r C_{\text{air}}\)

• Series combination: \(\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots\)

• Parallel combination: \(C_{\text{eq}} = C_1 + C_2 + \ldots\)

• Energy stored: \(U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}\)

• Energy density: \(u = \frac{1}{2}\epsilon E^2\)

Important Formulas - Quick Reference

• Coulomb’s Law: \(\mathbf{F} = \frac{q_1 q_2}{4\pi\epsilon_0 r^2} \hat{r}\)

• Electric Field: \(\mathbf{E} = \frac{q}{4\pi\epsilon_0 r^2} \hat{r}\), \(\mathbf{E} = -\nabla V\)

• Gauss’s Law: \(\oint \mathbf{D} \cdot d\mathbf{S} = Q_{\text{enc}}\), \(\nabla \cdot \mathbf{D} = \rho\)

• Potential: \(V = -\int \mathbf{E} \cdot d\mathbf{l}\)

• Capacitance: \(C = \frac{Q}{V}\)

• Dielectric: \(\mathbf{D} = \epsilon \mathbf{E}\), \(\mathbf{P} = \epsilon_0(\epsilon_r-1)\mathbf{E}\)

• Energy: \(U = \frac{1}{2}CV^2\), \(u = \frac{1}{2}\epsilon E^2\)

GATE Exam Strategy

Summary

• Coulomb’s Law: Foundation for all electrostatic problems

• Electric Field & Potential: Key for charge distributions

• Gauss’s Law: Powerful tool for symmetric problems

• Divergence: Links field to charge density

• Dielectrics: Modify fields and increase capacitance

• Capacitance: Essential for energy storage problems