Electromagnetic Fields and Waves - EENG388 Summer 2024

Overview of Electromagnetics

• Electromagnetics: Study of the effects of charges at rest and in motion.

• Special Cases of Electromagnetics:

  • Electrostatics: Charges at rest.

  • Magnetostatics: Charges in steady motion (DC).

  • Electromagnetic Waves: Waves generated by charges in time-varying motion.

Introduction to Electrostatics

• Electrostatic Fields: Easier to visualize compared to static magnetic fields; simpler than time-varying electromagnetic fields.

• Practical Applications:

  • Electronic devices operation control.

  • CCD imaging cameras.

  • Liquid crystal displays.

• Introduces coordinate systems, vectors, and vector algebra.

Fundamental Concepts in Electrostatics

• Coulomb's Law: Force between two charges Q1 and Q2 separated by a distance vector R12.

  • Formula: F_{12} = \frac{1}{4 \pi \epsilon_0} \frac{Q_1 Q_2}{R_{12}^2} \hat{a}_{12}

  • Force characteristics:

    • Repulsive: Like charges (both positive or both negative).

    • Attractive: Opposite charges.

Example Calculation Using Coulomb's Law (Example 2.2)

• Given:

  • Charge Q1 = 10 nC at (0.0, 0.0, 4.0 m)

  • Charge Q2 = 2.0 nC at (0.0, 4.0 m, 0.0)

• Calculate R12:

  • Vector from Q1 to Q2: R_{12} = 4a_y - 4a_z

  • Magnitude: |R_{12}| = \sqrt{ (0^2 + 4^2 + 4^2) } = 4 m

Electric Field Intensity

• Defined as the force per unit charge:

  • Formula: E_1 = \frac{F}{Q_2}

• Units: Volt/meter (V/m).

Example of Electric Field Intensity (Example 2.4)

• Find electric field intensity based on previous calculations.

Visualization of Electric Fields using Field Lines

• Field lines help visualize electric fields, following the direction of field vectors.

• Example: Field lines for a pair of opposite charges.

Spherical Coordinate System

• Used to describe electric field intensity from a point charge:

  • R = \sqrt{x^2 + y^2 + z^2}

• Fields are radially directed; magnitude depends solely on radial distance.

Electric Fields from Continuous Charge Distributions

• Types of Charge Distributions:

  • Line Charge: Charge density ( \rho_L ) (C/m).

  • Surface Charge: Charge density ( \rho_S ) (C/m^2).

  • Volume Charge: Charge density ( \rho_V ) (C/m^3).

• Calculating electric fields from these distributions involves integrals of their respective charge densities.

Electric Field from Infinite Length Line Charge

• Derivation Steps:

  1. Choose arbitrary observation point P.

  2. Use symmetry; field intensity varies only with radial distance.

  3. Integral calculations lead to the final form of electric field intensity.

Gauss’s Law

• States that the net electric flux through any closed surface is proportional to the total charge enclosed:

  • Formula: \Phi_E = \oint D \cdot dS = Q_{enc}

• Point Form of Gauss's Law:

  • 
    abla \cdot D = \rho_V

Electric Flux Density

• Electric flux density D reflects how many field lines pass through an area.

• Formula for electric flux through a surface:

  • \Phi = \int D , dS

• No flux comes through the cylindrical surface sides or end caps in certain conditions.

Electric Potential

• Work done in moving a charge from point a to b against an electric field:

  • V_{ab} = -\int E , dL

• Connection of electric field intensity to potential is expressed through gradients:

  • E = -
    abla V

• Application of Gauss’s Law helps compute electric fields and potentials in systems with spherical or cylindrical symmetry.

Summary of Key Concepts

1. Coulomb's Law & Electric Forces.

2. Definitions of electric field and potential.

3. Gauss’s Law and its applications to different charge distributions.

4. Importance of coordinate systems in electrostatics.

5. Electric field intensity and its relationship with electric potential.