Unlocking the Power of Electromagnetic Fields

Explore the four types of charge distribution—point, volume, line, and surface—and learn their densities rho_V, rho_L, and rho_S, and how Q is computed via volume, line, and surface integrations.

Explain electric field intensity from a point charge using Coulomb's law, deriving E = (1/4πε0) Q/d^2 in the direction from source to field point, and sum contributions for multiple charges.

Derives the electric field intensity of an infinite sheet with surface charge density rho, showing E = rho/(2 epsilon0) z-hat perpendicular to the sheet via differential surface charges and integration.

Derive the electric field intensity of an infinite line charge using a differential length along the z-axis, cylindrical coordinates, and the foot-of-perpendicular distance, yielding a general radially directed field expression.

Explain electric flux and electric flux density, show charge displacement between concentric spheres, and derive D = ε0 E (and D = ε0 ε_r E) for media.

Apply Gauss's law to relate electric flux through a closed surface to the enclosed charge, using differential surface elements and the surface integral of the electric flux density.

Derive the divergence of electric flux density D using a small rectangular box and closed-surface integration, then take the limit to obtain ∇·D = ∂Dx/∂x + ∂Dy/∂y + ∂Dz/∂z.

Explore Maxwell's first equation in electrostatics by applying Gauss's law, showing that the volume charge density equals the divergence of the electric flux density D.

Derive the divergence theorem from Maxwell's first equation in electromagnetism, linking the closed surface integral of D to the volume integral of ρ.

Explain how to compute potential difference in a point charge field via work against the field and derive V(r) = Q/(4π ε0 r) with infinity as reference.

Explore the concept of potential gradient, rate of change of potential with distance, and how the electric field equals the negative gradient of potential in Cartesian, cylindrical, and spherical coordinates.

The lecture defines an electric dipole as opposite-sign charges +Q and -Q, derives exact potential and far-field form V ≈ p cos theta /(4π ε0 r^2) with p = QD.

Explain energy density as the average work per unit volume in an electrostatic field, derived from transferring charges into and out of a volume using W = QV.

Define energy density in an electrostatic field and convert charge-based expressions to volume integrals, deriving three equivalent forms for the average energy density, including 1/2 ε0 |E|^2.

Derive the continuity equation of current by connecting current density and charge density via divergence. Learn how surface and volume integration lead to ∇·J = -∂ρ/∂t.

Derives boundary conditions at a conductor free-space boundary: tangential E and D vanish, while normal D and E relate to surface charge density, Dn = rho_s, En = rho_s/epsilon0.

Explore boundary conditions at the interface of two dielectrics. Show E_t1 = E_t2 and D_n1 = D_n2, with D_t = epsilon0 epsilon_r E_t and E_n1 related to E_n2 via epsilon_r2/epsilon_r1.

Explore conductor–dielectric boundary conditions, deriving tangential electric field and flux density are zero at the boundary, while the normal components satisfy D_n = rho_s and E_n = rho_s/(epsilon0 epsilon_r).

Explore the concept of capacitance and derive the parallel plate capacitor formula, showing how charge, voltage, dielectric permittivity, plate area, and separation determine the capacitance.

Explore Poisson and Laplace equations and their applications in astronomy, heat flow, fluid dynamics, and electromagnetism, then derive their relations with electric field, potential, and permittivity.

This lecture derives the cylindrical coaxial capacitor's capacitance using concentric cylinders of radii A and B and a dielectric, giving C = 2π ε0 εr L / ln(B/A).

Derives the general capacitance of a spherical capacitor with inner radius a and outer radius b. Then analyzes an isolated sphere and a dielectric-coated inner sphere with infinite outer radius.

Examine current distributions in various conductors: cylindrical cross sections with current density J, filament and rectangular slab conductors, and strip conductors with surface current density K.

Compute electric field magnitude from the electrostatic potential in Cartesian coordinates by applying the gradient to phi = 2 x sqrt(y) and evaluating at x = 1, y = 1.

Apply the divergence formula to a given electrostatic vector field in Cartesian coordinates, computing ∂Vx/∂x + ∂Vy/∂y + ∂Vz/∂z and converting unit vectors to determine the correct option.

Explore the Biot-Savart law, linking current in a filament to magnetic field intensity at a point using differential length, distance, and angle.

Derive the magnetic field intensity of an infinite long filament conductor using Biot-Savart on a differential length dl and integrate in cylindrical coordinates.

Learn Ampere's circuital law: the line integral of H around a closed path equals the current enclosed. Compare it with Biot–Savart law and explore applications to electromagnets, motors, and transformers.

Using the image charge method, a charge at (0,0,2) near a conducting plane produces a total electrostatic field at (√2,√2,0) of -2 k̂ N/C.

Explore composite capacitors formed by two dielectrics between conductors, and show how the total capacitance arises as the series combination of C1 and C2.

Explore two dielectric materials between parallel plates forming a composite capacitor in parallel, derive C = C1 + C2, and apply areas A1 and A2 with εr1 and εr2.

Apply Ampere's circuital law to determine the magnetic field intensity in a coaxial cable, analyzing inner and outer conductors, dielectric, and current enclosed.

Apply ampere's circuital law to a coaxial cable to compute magnetic field intensity across multiple paths. Analyze inner and outer conductor currents and derive h-bar for each region.

Compute the electric field intensity at (0,1,0) due to a 1 μC point charge at (−1,1,1), yielding E = (2 i - k) × 10^-6 /(20 sqrt(5) pi epsilon0) V/m.

An analysis of a three-dielectric composite capacitor between square plates, with relative permittivities 1, 2, and 4; two capacitors in series in parallel with the third to yield 4.72 pF.

derive the electric field from the given potential, compute its magnitude at point 1 -1 1 as 100√3 V/m, and state its direction as (-i + j - k)/√3.

Explore the point form of Ampere's law and the curl of magnetic field intensity. Del bar cross H equals J links line integrals around a closed path to surface area.

Explore the curl of the magnetic field intensity and Ampere's circuital law, deriving del × H = J using Cartesian, cylindrical, and spherical coordinates and practical path analyses.

The lecture derives the Lorentz force equation, showing a moving charge experiences F = q(E + v × B), combining electric and magnetic forces.

Explore the application of the Lorentz force equation to differential current elements, including slip-type, slab, and strip conductors, deriving forces via I dℓ × B.

Explore Stokes theorem, linking the surface integral of curl of a vector field to the line integral around its boundary, with an electromagnetic form using magnetic field intensity H.

Explore magnetic boundary conditions between two materials with different relative permeabilities. Derive normal B continuity and tangential H relations using Gauss’s and Ampere’s laws, including surface current K.

Explore Maxwell's equations for time varying fields, focusing on integral and differential forms and Faraday's law, which links emf to the rate of change of magnetic flux.

Derives Maxwell's equations for time varying fields from Ampere's circuital law, detailing differential and integral forms, including displacement current density and the continuity equation.

Explains time-varying Maxwell equations by deriving the second equation in differential form, introducing displacement current J_d, and presenting its integral form via Ampere's law and Stokes' theorem.

The lecture analyzes a parallel plate capacitor with air dielectric, then a half-area dielectric insert of relative permittivity ε_r, showing the equivalent capacitance C = C0(1+ε_r)/2 for figure B.

In this lecture, a three-dielectric parallel-plate capacitor with a symmetrical middle layer is analyzed; using the 2-volt near-interface drops, it yields epsilon1:epsilon2 = 3:2.

Analyze an electrostatic field problem on a parallel-plate capacitor with a half-thickness dielectric. Determine the relative permittivity ε_r from capacitance change from 60 to 86 pF, yielding ε_r = 2.53.

Two semi-infinite conducting sheets at right angles create image charges for a plus q at distance d, enabling a Coulomb-based net force f = q^2/(4 pi epsilon0) * K / d^2.

Explore a numerical problem in electrostatics at a plane boundary between two semi-infinite dielectrics with ε_r1=2 and ε_r2=5, using boundary conditions to relate E across the interface.

This lecture derives energy stored in magnetic field from the electric-field framework, introducing H bar and B bar and outlining volume-integral methods for magnetic energy and energy density.