Magnetostatics: Quick Notes for GATE Exam

Electrostatics (Brief Review)

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: \(C = \frac{\epsilon_0 A}{d}\)

Magnetostatics

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

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

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 Problem-Solving Tips

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