Course Overview

  • Course: Engineering Electromagnetics

  • Instructor: Mohamed Fareq Malek

  • University: UOWD

  • Focus: Lecture 3 discusses Capacitance, Poisson’s Equations, Steady Magnetic Field, and Maxwell’s Equations for Static Fields.

References

  • Primary Reference: W.H. Hayt Jr. and J. A. Buck

  • Chapters covered: 6, 7, 8

  • Outline covered in the lecture:

    • Capacitance

    • Poisson’s Equations

    • Steady Magnetic Field

    • Maxwell’s Equations for Static Fields

3.1 Capacitance

  • Definition: Capacitance quantifies the ability of a device to store electrical energy.

  • Basic Structure of a Capacitor:

    • Comprises two oppositely charged conductors separated by a dielectric material.

  • Capacitance Formula:

    • C = \frac{Q}{V_0} where:

    • C = Capacitance (in Farads)

    • Q = Stored Charge

    • V_0 = Applied Voltage

  • Stored Energy in a Capacitor:

    • Energy stored is dependent on capacitance and voltage.

Steps to Calculate Capacitance (D. Cheng)

  1. Select a suitable coordinate system based on the geometry of the capacitor.

  2. Assume charges +Q and -Q on conductors.

  3. Calculate the electric field (E).

  4. Evaluate the potential difference (V_0).

  5. Determine the capacitance ratio.

  • Calculation example provided:

    • For a coaxial transmission line:

    • Assumes hollow inner conductor with equal and opposite charges on inner and outer conductors.

    • The electric field E is zero elsewhere.

    • Charge density on inner conductor is used to derive capacitance.

3.2 Poisson's Equations

  • Starting Point: Derived from Maxwell’s First Equation.

  • Formulation of Poisson’s Equation:

    • \nabla^2 V = -\frac{\rho}{\epsilon}

  • Purpose: Solves for potential field in a region when the electric field or potential values at boundaries are known.

Mathematical Concepts

  • Laplacian Operator:

    • Denoted as \nabla^2 (pronounced “del squared”). It represents the divergence of the gradient of a function.

  • In Cases of Zero Charge Density: Poisson’s equation simplifies to Laplace’s equation:

    • \nabla^2 V = 0.

Example: Parallel Plate Capacitor

  • Assumption: Plate separation (d) much smaller than plate dimensions.

  • Laplace’s equation changes based on this simplification; after integration, boundary conditions yield:

    • V(x=0) = 0,

    • V(x=d) = V_0.

3.3 Steady Magnetic Field

  • Definition: Magnetic Field (H) generated by currents in circuits.

  • Biot-Savart Law: Predicts magnetic field intensity (H) based on a point source current element (dL):

    • [H] = A/m.

  • Key Concept: Models forces between currents effectively regardless of their orientations.

Definitions & Applications

  • Magnetic Flux Density: Measures the magnetic field strength, expressed in Weber/m² or Tesla.

Ampere’s Circuital Law

  • Statement: The line integral of H around a closed path equals the current I enclosed.

  • Independent of material properties and path choice.

  • Application: For cylindrical coordinates around an infinite wire, we expect magnetic field intensity H to vary only with the radial distance (r).

Example: Coaxial Transmission Line

  • Configuration: Two concentric conductors carrying equal and opposite currents (I).

  • Goal: Find magnetic field for all radial distances (r).

  • Expressions derived for different radial regions:

    • For 0 < r < a: Field decreased by a factor of radius,

    • For a < r < b: Field proportional to the total current,

    • For r > c, total current enclosed is zero.

3.4 Maxwell's Equations for Static Fields

  • Context: Foundational theory underlying electrostatics and magnetostatics.

Mathematical Concepts

  • Divergence and Curl:

    • Divergence indicates the source strength of an electric field, and curl describes circulation strength of a magnetic field.

  • Curl Definition: Describes the rotation of a vector field.

    • Given by:
      \text{Curl} \vec{F} = \nabla \times \vec{F}

  • Computation of Curl: In rectangular coordinates, it involves cross products.

Coordinate System Formulation

  • Cylindrical Coordinates: Describes transformation of vector fields based on different geometric configurations.

  • Spherical Coordinates: Similar to cylindrical but adapted for spherical symmetry.

Notable Examples

  • Provided Example for Currents: Calculated curl in relation to a given magnetic field.

  • Maxwell’s Equations Final Form: For static fields in point form:

    1. Gauss’ Law for electric fields,

    2. Conservative property of the static electric field,

    3. Ampere’s Circuital Law,

    4. Gauss' Law for magnetic fields.

  • Integration over regions yields various integral forms, extending implications for dynamics and time-varying fields in future discussions.