Electrostatics - Key Formulas

• Coulomb’s Law: \(\mathbf{F} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{r}\)

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

• Gauss’s Law: \(\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0}\)

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

• Capacitance: \(C = \frac{Q}{V}\), for parallel plate: \(C = \frac{\epsilon_0 A}{d}\)

Biot-Savart’s Law

• Fundamental Law: Magnetic field due to current element:

  \[d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \hat{r}}{r^2}\]

• Key Points:

  • Direction: Right-hand rule

  • Magnitude: \(|d\mathbf{B}| = \frac{\mu_0 I dl \sin\theta}{4\pi r^2}\)

  • Units: Tesla (T)

• Applications: Straight wire, circular loop, finite wire segments

Biot-Savart’s Law - Important Results

• Infinite straight wire:

  \[B = \frac{\mu_0 I}{2\pi r}\]

• Circular loop at center:

  \[B = \frac{\mu_0 I}{2R}\]

• Circular loop on axis:

  \[B = \frac{\mu_0 I R^2}{2(R^2 + z^2)^{3/2}}\]

• Finite wire: Complex integral - use standard results

Ampere’s Law

• Statement: Line integral of magnetic field equals enclosed current:

  \[\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}}\]

• Conditions for Application:

  • High degree of symmetry

  • Closed Amperian loop

  • Constant \(|\mathbf{B}|\) along chosen path

• Sign Convention: Current direction by right-hand rule

Ampere’s Law - Applications

• Solenoid (inside):

  \[B = \mu_0 n I = \mu_0 \frac{N}{l} I\]
  where \(n\) is turns per unit length

• Toroid:

  \[B = \frac{\mu_0 N I}{2\pi r}\]

• Coaxial cable: Between conductors

• Current sheet: \(B = \frac{\mu_0 K}{2}\) where \(K\) is surface current density

Curl of Magnetic Field

• Definition: Differential form of Ampere’s law:

  \[\nabla \times \mathbf{B} = \mu_0 \mathbf{J}\]

• Physical Meaning: Circulation of \(\mathbf{B}\) per unit area

• Cartesian Form:

  \[\nabla \times \mathbf{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ B_x & B_y & B_z \end{vmatrix}\]

• Important: \(\nabla \times \mathbf{B} = 0\) in current-free regions

Curl in Other Coordinate Systems

• Cylindrical:

  \[\nabla \times \mathbf{B} = \left(\frac{1}{\rho}\frac{\partial B_z}{\partial \phi} - \frac{\partial B_\phi}{\partial z}\right)\hat{\rho} + \left(\frac{\partial B_\rho}{\partial z} - \frac{\partial B_z}{\partial \rho}\right)\hat{\phi} + \frac{1}{\rho}\left(\frac{\partial(\rho B_\phi)}{\partial \rho} - \frac{\partial B_\rho}{\partial \phi}\right)\hat{z}\]

• Spherical:

  \[\nabla \times \mathbf{B} = \frac{1}{r\sin\theta}\left(\frac{\partial B_\phi}{\partial \theta} - \frac{\partial(\sin\theta B_\theta)}{\partial \phi}\right)\hat{r} + \cdots\]

• GATE Tip: Use symmetry to simplify calculations

Magnetomotive Force (MMF)

• Definition: Driving force for magnetic flux

  \[\text{MMF} = N I \text{ (Ampere-turns)}\]

• Physical Analogy: Like EMF in electric circuits

• Applications:

  • Solenoids: MMF = Number of turns × Current

  • Transformers: Primary and secondary MMF

  • Electric machines: Stator and rotor MMF

• Unit: Ampere-turns (AT)

Reluctance

• Definition: Opposition to magnetic flux

  \[\mathcal{R} = \frac{l}{\mu A} = \frac{l}{\mu_0 \mu_r A}\]

• Parameters:

  • \(l\): Length of magnetic path

  • \(\mu\): Permeability (\(\mu = \mu_0 \mu_r\))

  • \(A\): Cross-sectional area

• Unit: Henry\(^{-1}\) (H\(^{-1}\))

• Analogy: Resistance in electric circuits

Reluctance - Key Points

• Series Reluctances: \(\mathcal{R}_{\text{total}} = \mathcal{R}_1 + \mathcal{R}_2 + \cdots\)

• Parallel Reluctances: \(\frac{1}{\mathcal{R}_{\text{total}}} = \frac{1}{\mathcal{R}_1} + \frac{1}{\mathcal{R}_2} + \cdots\)

• Air Gap Effect: High reluctance due to \(\mu_r = 1\)

• Ferromagnetic Materials: Low reluctance due to high \(\mu_r\)

• GATE Tip: Most flux passes through low reluctance path

Magnetic Circuits

• Ohm’s Law for Magnetic Circuits:

  \[\Phi = \frac{\text{MMF}}{\mathcal{R}} = \frac{NI}{\mathcal{R}}\]

• Analogies with Electric Circuits:

  Electric	Magnetic
  Current (I)	Flux (\(\Phi\))
  Voltage (V)	MMF (NI)
  Resistance (R)	Reluctance (\(\mathcal{R}\))
  Conductivity	Permeability

Magnetic Circuits - Analysis Steps

1. Identify: Magnetic path and cross-sections

2. Calculate: Reluctance of each section

   \[\mathcal{R}_i = \frac{l_i}{\mu_i A_i}\]

3. Find: Total reluctance (series/parallel combination)

4. Apply: Ohm’s law for magnetic circuits

   \[\Phi = \frac{\text{MMF}}{\mathcal{R}_{\text{total}}}\]

5. Calculate: Flux density \(B = \frac{\Phi}{A}\)

Magnetic Circuits - Applications

• Transformers:

  • Core reluctance

  • Air gap effects

  • Leakage flux

• Inductors:

  • Energy storage

  • Inductance calculation

• Electric Machines:

  • Motor and generator analysis

  • Magnetic field distribution

Maxwell’s Equations - Complete Set

• Gauss’s Law (Electric):

  \[\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \quad \text{or} \quad \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\epsilon_0}\]

• Gauss’s Law (Magnetic):

  \[\nabla \cdot \mathbf{B} = 0 \quad \text{or} \quad \oint \mathbf{B} \cdot d\mathbf{A} = 0\]

• Faraday’s Law:

  \[\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \quad \text{or} \quad \oint \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}\]

• Ampere-Maxwell Law:

  \[\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t}\]

Boundary Conditions for EM Fields

• Electric Field:

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

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

• Magnetic Field:

  • Normal component: \(B_{1n} = B_{2n}\)

  • Tangential component: \(H_{1t} - H_{2t} = J_s\)

• GATE Applications: Dielectric interfaces, conductor boundaries

GATE Strategy for Magnetostatics

• Identify Symmetry: Choose appropriate law (Biot-Savart vs Ampere)

• Current Configurations:

  • Straight wire \(\to\) Biot-Savart or Ampere

  • Circular loop \(\to\) Biot-Savart

  • Solenoid/Toroid \(\to\) Ampere’s law

• Magnetic Circuits: Draw equivalent circuit, calculate reluctances

• Common Mistakes: Sign errors, wrong coordinate systems

• Units: Always check dimensional consistency

Summary - Key Points to Remember

• Biot-Savart’s Law: For any current configuration

• Ampere’s Law: For symmetric configurations only

• Curl: \(\nabla \times \mathbf{B} = \mu_0 \mathbf{J}\) in magnetostatics

• MMF: Driving force for magnetic flux (NI)

• Reluctance: Opposition to flux \(\left(\frac{l}{\mu A}\right)\)

• Magnetic Circuits: Use electrical circuit analogy

• Practice: Solve problems with different geometries

Focus on understanding physical concepts and mathematical relationships!

Final Tips for GATE Success

• Master the Fundamentals: Understand derivations, not just formulas

• Practice Numerical Problems: Time management is crucial

• Vector Calculus: Strong foundation in div, curl, gradient

• Coordinate Systems: Comfortable with Cartesian, cylindrical, spherical

• Symmetry Arguments: Often simplify complex problems