Coulomb’s Law
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Force between two point charges:
\[\mathbf{F} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{r} = k \frac{q_1 q_2}{r^2} \hat{r}\] -
\(k = \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \, \text{N.m^2/C^2}\)
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\(\epsilon_0 = 8.854 \times 10^{-12} \, \text{F/m}\) (permittivity of free space)
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Key Points:
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Inverse square law
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Like charges repel, unlike charges attract
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Superposition principle applies
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Electric Field Intensity
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Electric field due to a point charge:
\[\mathbf{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}\] -
Superposition principle: \(\mathbf{E}_{\text{total}} = \sum \mathbf{E}_i\)
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Units: N/C or V/m
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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
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Defined as: \(\mathbf{D} = \epsilon_0 \mathbf{E}\) (in vacuum)
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In dielectric medium: \(\mathbf{D} = \epsilon \mathbf{E} = \epsilon_0 \epsilon_r \mathbf{E}\)
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Units: C/m2
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Key Properties:
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Independent of medium properties
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Continuous across dielectric boundaries
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Useful in Gauss’s Law applications
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Electric flux: \(\Phi = \int \mathbf{D} \cdot d\mathbf{S}\)
Gauss’s Law
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Integral form: \(\oint \mathbf{D} \cdot d\mathbf{S} = Q_{\text{enc}}\)
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Differential form: \(\nabla \cdot \mathbf{D} = \rho\)
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For vacuum: \(\oint \mathbf{E} \cdot d\mathbf{S} = \frac{Q_{\text{enc}}}{\epsilon_0}\)
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Applications:
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Spherical symmetry: Choose spherical Gaussian surface
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Cylindrical symmetry: Choose cylindrical Gaussian surface
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Planar symmetry: Choose box-shaped Gaussian surface
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GATE Tip: Always exploit symmetry!
Divergence
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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
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Key Point: Positive divergence = source, Negative = sink
Electric Field and Potential
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Potential: \(V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}\)
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Electric field: \(\mathbf{E} = -\nabla V = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}\)
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For multiple point charges:
\[V = \frac{1}{4\pi\epsilon_0} \sum_i \frac{q_i}{r_i}\] -
Key Relations:
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\(\mathbf{E} = -\nabla V\)
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\(V = -\int \mathbf{E} \cdot d\mathbf{l}\)
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Work done: \(W = q(V_f - V_i)\)
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Linear charge density: \(\lambda = \frac{dq}{dl}\)
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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!
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Surface charge density: \(\sigma = \frac{dq}{dS}\)
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For infinite plane sheet:
\[\mathbf{E} = \frac{\sigma}{2\epsilon_0} \hat{n}\] -
Field is uniform and perpendicular to plane
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Potential: \(V = -\frac{\sigma}{2\epsilon_0} |z|\) (for sheet at z=0)
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Two parallel plates:
\[\mathbf{E} = \frac{\sigma}{\epsilon_0} \hat{n} \quad \text{(between plates)}\]
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Volume charge density: \(\rho = \frac{dq}{dV}\)
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For uniform sphere of radius \(a\):
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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
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Dielectric reduces electric field:
\[\mathbf{D} = \epsilon \mathbf{E} = \epsilon_0 \epsilon_r \mathbf{E}\] -
\(\epsilon_r\): Relative permittivity (dielectric constant)
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Polarization vector: \(\mathbf{P} = \epsilon_0 (\epsilon_r - 1) \mathbf{E}\)
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Relationship: \(\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}\)
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Boundary Conditions:
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Normal component: \(D_{1n} - D_{2n} = \sigma_s\)
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Tangential component: \(E_{1t} - E_{2t} = 0\)
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No free surface charge (\(\sigma_s = 0\)):
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\(D_{1n} = D_{2n}\) (normal D continuous)
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\(E_{1t} = E_{2t}\) (tangential E continuous)
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Refraction of field lines:
\[\frac{\tan \theta_1}{\tan \theta_2} = \frac{\epsilon_1}{\epsilon_2}\] -
GATE Application: Capacitor with multiple dielectrics
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Fields in different media related by permittivity ratios
Capacitance
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Definition: \(C = \frac{Q}{V}\), Units: F (Farad)
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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 \(b\))
\[C = 4\pi\epsilon \frac{ab}{b-a}\]
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Isolated conducting sphere: \(C = 4\pi\epsilon a\)
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Effect of dielectric: \(C_{\text{new}} = \epsilon_r C_{\text{air}}\)
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Series combination: \(\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots\)
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Parallel combination: \(C_{\text{eq}} = C_1 + C_2 + \ldots\)
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Energy stored: \(U = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{Q^2}{2C}\)
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Energy density: \(u = \frac{1}{2}\epsilon E^2\)
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Coulomb’s Law: \(\mathbf{F} = \frac{q_1 q_2}{4\pi\epsilon_0 r^2} \hat{r}\)
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Electric Field: \(\mathbf{E} = \frac{q}{4\pi\epsilon_0 r^2} \hat{r}\), \(\mathbf{E} = -\nabla V\)
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Gauss’s Law: \(\oint \mathbf{D} \cdot d\mathbf{S} = Q_{\text{enc}}\), \(\nabla \cdot \mathbf{D} = \rho\)
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Potential: \(V = -\int \mathbf{E} \cdot d\mathbf{l}\)
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Capacitance: \(C = \frac{Q}{V}\)
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Dielectric: \(\mathbf{D} = \epsilon \mathbf{E}\), \(\mathbf{P} = \epsilon_0(\epsilon_r-1)\mathbf{E}\)
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Energy: \(U = \frac{1}{2}CV^2\), \(u = \frac{1}{2}\epsilon E^2\)
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Symmetry is your friend: Always look for spherical, cylindrical, or planar symmetry
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Boundary conditions: Master dielectric interface problems
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Superposition: Use for multiple charge configurations
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Units & Dimensions: Double-check your final answers
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Common mistakes: Sign errors, wrong reference points for potential
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Practice: Solve previous year questions extensively
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Coulomb’s Law: Foundation for all electrostatic problems
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Electric Field & Potential: Key for charge distributions
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Gauss’s Law: Powerful tool for symmetric problems
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Divergence: Links field to charge density
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Dielectrics: Modify fields and increase capacitance
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Capacitance: Essential for energy storage problems