Electrostatics: Quick Notes for GATE Exam

Coulomb’s Law

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 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

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

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

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

Electric Field and Potential

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

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

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

Key Tips for GATE Success

  • Symmetry is your friend: Always look for spherical, cylindrical, or planar symmetry

  • Boundary conditions: Master dielectric interface problems

  • Superposition: Use for multiple charge configurations

  • Units & Dimensions: Double-check your final answers

  • Common mistakes: Sign errors, wrong reference points for potential

  • Practice: Solve previous year questions extensively

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