Electromagnetic Fields and Waves

Compute dot products for parallel and perpendicular vectors using cos 0 and cos 90 degrees, applying commutative law. Identify unit vectors in an orthogonal coordinate system, where self-dot equals one.

Explore vector algebra and vector calculus, covering scalar properties, scaling, addition and subtraction, coplanar vectors, and the parallelogram rule with commutative, associative, and distributive properties.

Compute the angle between vectors using the dot product formula, solving cos theta = (A dot B)/(|A||B|) and applying it to the example to find theta in degrees.

Demonstrate the corkscrew rule and its right-hand and left-hand variants for A cross B, with magnitude |A||B| sin theta and a resultant perpendicular to the surface.

Explore cross product properties, including non-commutativity, and express A × B as a determinant; apply scalar and vector triple products and the bac-cab rule to cyclic permutations.

Explore rectangular (Cartesian) coordinate systems and compare them with cylindrical and spherical coordinates, showing how each assigns a point's address in space.

Review coordinate systems: Cartesian (rectangular), cylindrical, and spherical. Derive differential length, surface, and volume using dx, dy, dz and unit vectors x̂, ŷ, ẑ.

Understand the cylindrical coordinate system with rho, phi, and z. Derive differential length dl = drho a_rho + rho dphi a_phi + dz a_z and arc length rho dphi.

Compute the enclosed charge in a unit cube with density rho = 2 x^2 y by evaluating a triple integral, yielding Q = 1/3.

Explains the spherical coordinate system for a ball, with points r theta phi and ranges 0≤r≤1, 0≤theta≤pi, 0≤phi≤2pi, and derives dl and dV using ar, atheta, aphi.

Learn rectangular to cylindrical transformations: x = rho cos phi, y = rho sin phi, z = z, with rho = sqrt(x^2+y^2) and phi = arctan(y/x).

Learn to perform cylindrical to spherical and rectangular to spherical point transformations, deriving rho, theta, phi relations with x, y, z for accurate coordinate conversions.

the del operator is a vector operator in cartesian coordinates that serves as the gradient operator for scalars and enables divergence, curl, and the Laplacian of fields.

Explore divergence as field lines spreading and flux changes (positive, negative, zero), and curl as rotation around a closed path, with rotational versus irrotational fields.

Learn how the del operator acts on gradient to yield divergence and curl, and understand the gradient as the maximum spatial rate of change using a temperature furnace example.

Explore divergence as net outward flux per unit volume and curl as the maximum circulation. Relate them via Stokes' theorem: circulation around a closed path equals surface integral of curl.

Explore electrostatics and Coulomb's law, where the force between two charges is proportional to Q1 Q2 and inversely proportional to distance squared, F equals k Q1 Q2 over R squared.

Explore Coulomb's law in vector form and learn to compute the net force on multiple charges using vector addition and position vectors.

Apply coulomb's law to N charges by summing vector forces F_k = ke q q_k (R − r_k)/|R − r_k|^3 to obtain the total force.

Defines electric field intensity as the force per unit charge, derives the total field from multiple point charges, and explains unit test charge and radial field lines with equipotential regions.

Explore point, line, surface, and volume charge distributions and how Coulomb's law applies to each, with examples like filament in a bulb and a parallel plate capacitor illustrating charge enclosed.

Explore the electric field intensity from a line element, use symmetry to derive the finite line element, and obtain E = rho/(2 pi epsilon naught) for a straight conductor.

explain electric field intensity at a point due to a circular ring of charge, show symmetry cancels radial components, leaving the axial component whose magnitude depends on rho and z.

the lecture shows that an infinite sheet of charge produces a perpendicular electric field equal to rho_s/(2 ε0) along the sheet normal, with horizontal components canceling by symmetry.

Explore electric flux as the total flux lines in a field. Flux lines originate at positive charges, terminate at negative charges, never cross, and more lines imply a stronger field.

analyze electric flux density and its link to electric field for point, line, sheet, and volume charges, with D and E formulas for each distribution.

Apply Gauss law to relate the outward flux through a closed surface to the enclosed charge, and derive del dot D = rho v as Maxwell's first equation.

Gauss's law applies to symmetric charge distributions on a closed Gaussian surface and relates E and D fields, considering whether D is normal or tangential to the surface.

Demonstrate equivalent gaussian surfaces for point, line, and surface charges using spherical, cylindrical, and cubic surfaces to model flux and d bar.

Gauss's law applications cover point, infinite line, surface, and volume charges. For a point charge, use a spherical Gaussian surface; the radial flux leads to D = Q/(4π r^2) rhat.

Apply Gauss's law to an infinite line charge along the z-axis using a cylindrical Gaussian surface. Reveal that the radial flux density Dρ = λ/(2πρ) and Q = λL.

Apply Gauss's law to an infinite sheet of charge in the x-y plane and determine the electric flux density D at point p, with D directed normal to the sheet.

Apply Gauss's law to a Uniformly charged sphere, deriving D and E for r < a and r > a with a Gaussian surface and volume charge rho.

Understand energy conservation in electric fields, moving a test charge from infinity to a point, with work and voltage, which is path-independent and sums the potentials from charges.

learn how potential difference is defined and calculated from infinity, derive the potential V = Q/(4 pi epsilon0 r), and recognize path independence and sign conventions for charges in volts.

Compute the electric potential at point A from multiple point charges using vector notation and the superposition principle, arriving at V = sum q_i /(4 pi epsilon_naught |R - r_i|).

Explore Maxwell's second equation in integral and differential forms, linking E and V, showing the closed-path line integral of E vanishes and equals curl E = 0 via Stokes' theorem.

Demonstrate that the electric field equals minus the gradient of the potential by expanding del into x, y, z components and linking dv to e dot dl.

Explore the energy stored in an electrostatic field around point charges and energy density. Derive the total electrostatic energy as one-half the sum of qi times the potential.

Learn how static electrostatic energy extends from point charges to volume, line, and surface charges, yielding W_e = 1/2 ε0 ∫ E^2 dv.

Explore the continuity equation for time varying fields, derived from charge conservation. Relate the time rate of change of charge density to the divergence of current density, via divergence theorem.

Defines relaxation time as the time for interior charge to drop to 36.8% of its initial value, deriving the exponential decay using Maxwell's equations and ε and σ.

Learn how Poisson's equation links electric potential to charge density and how Laplace's equation applies in regions, with E = -∇V, and explore Cartesian, cylindrical, and spherical forms.

Explore magnetostatics, comparing it to electrostatics, and learn how moving charges produce magnetic fields and lines from north to south, with the Biot-Savart law.

Biot–Savart law gives the differential magnetic field intensity H at a point from a current element, via dl × r over 4π r^2, with K and J extensions.

Using biot-savart, h from a line current; finite lines give h = I/(2 pi rho) a_phi_bar, infinite lines give h = I/(4 pi rho) (sin alpha2 - sin alpha1) a_phi_bar.

Derive the magnetic field intensity h bar on the axis of a circular current loop using the Biot-Savart law. Show axial symmetry and the on-axis field, including the center value.

Relate the line integral of the magnetic field intensity around a closed path to the net current enclosed, and derive curl H = J via Stokes' theorem as Ampere's law.

Demonstrate Ampere's circuital law by integrating the magnetic field intensity around a closed path for a line element, showing the integral equals the enclosed current.

The lecture derives the magnetic field intensity of an infinite sheet of current, showing H = (K × n)/2 with |H| = |K|/2, directed along ±x depending on z.

Understand magnetic flux and magnetic flux density, where flux is lines from north to south and density is flux per area, with B = mu H and no isolated poles.

The magnetic scalar potential VM exists in regions with zero current density, defined as the line integral of the magnetic field intensity; H = -∇VM and ∇^2 VM = 0.

Define the vector magnetic potential A and show that B = ∇×A, with A_d = μ0/4π ∫ Idl/r for line, surface, and volume elements, noting near- and far-field behavior.

Faraday's law states that induced emf equals minus the rate of change of magnetic flux linkages, scaled by the number of turns, for stationary loops, time-varying fields, or both.

Investigate transformer emf: a stationary loop in a time-varying magnetic field induces emf via Faraday’s law, linking e·dl, ∇×e, and dφ/dt with primary and secondary turns.

A moving loop in a stationary magnetic field experiences motional emf as it cuts the flux; emf equals the line integral of v cross B, for dc motors.

Explore how a moving loop in a time-varying magnetic field combines transformer emf and motional emf, linking E·dl to -dΦ/dt and v×B effects via Stokes' theorem.

Shows the inconsistency of Ampere's circuit law in electrostatic and time-varying fields, and derives curl H = J + displacement current density, linking to Maxwell's equations and modern technology.

Examine displacement current density in a parallel RC circuit, distinguishing conduction and displacement currents, deriving J and E relations, and noting how frequency shifts the medium between conductor and dielectric.

Learn boundary conditions at surfaces between media, such as dielectric–dielectric and conductor–dielectric, and decompose E and D into tangential and normal components using Maxwell’s equations.

Apply Maxwell's equations to dielectric-dielectric boundaries to show the tangential electric field is continuous and the normal displacement field D is continuous when free charge is absent.

At a conductor–dielectric boundary, the field inside a perfect conductor is zero and tangential E vanishes; the normal D equals the surface charge ρ_s, with D = ε0 ε_r E.

Learn magnetic boundary conditions at media interfaces using magnetic Gauss law and Ampere's law. Show normal fields equal across boundary and mu1 H1N equals mu2 H2N, with tangential H discontinuity.

Explore how an electromagnetic wave combines perpendicular electric and magnetic fields, propagates as a uniform plane wave, and reveals impedance and the e dot h product.

Derive the electromagnetic wave equations from Maxwell’s equations for time-varying fields, showing propagation in free space and dielectrics and del^2 E = mu0 epsilon0 ∂^2 E/∂t^2.

Derives the wave equation for electric and magnetic fields in a conducting medium, showing how conductivity and permittivity shape wave propagation through Maxwell's equations with J = σE.

Identify a plane wave as having constant phase on a plane surface. A uniform plane wave also has constant amplitude and travels in the x direction, with intrinsic impedance eta.

Derives the relation between E and H for a uniform plane wave in free space, showing intrinsic impedance eta equals E/H = sqrt(mu0/epsilon0) ≈ 377 ohms.

Examine sinusoidal time variations and the phasor form of the EM wave equation in a dielectric, deriving gamma = alpha + beta with alpha = 0 and beta = ω√(μ0ε0).

Derive the conducting medium wave equation in phasor form, establishing gamma squared equals j omega mu times (sigma plus j omega epsilon) and linking to the free-space case.

Explore Maxwell's equations in phasor form for time-varying fields, using t=0 and omega. Show phasor relations: ∇·E=ρ, ∇×E=-jωμH, ∇×H=(σ+jωε)E, and ∇·B=0 with D=εE and B=μH.

Derive the electric-field wave equation in a lossless medium and express the real field as a superposition of traveling waves in opposite directions using cos omega t ± beta x.

Explain wave propagation in a lossy conducting medium and the resulting attenuation, deriving the wave equation and gamma = alpha + j beta, with E = E0 e^{-alpha x} cos(ω t - beta x).

Derive the attenuation constant alpha and the phase constant beta for a lossy conducting medium from gamma = alpha + j beta, using omega, mu, sigma, and epsilon.

Explore how a good dielectric uses sigma/(omega epsilon) ≪ 1 and binomial expansion to derive alpha and beta for electromagnetic waves.

Derive alpha and beta for a good conductor as sqrt(omega mu sigma / 2); relate gamma, velocity of propagation, defined as omega / beta, and intrinsic impedance eta.