Electromagnetism Part-1

Define vectors as quantities with direction and magnitude, and apply triangular and parallelogram laws for addition. Explore dot products, projections, scalar multiplication, null vectors, and vector norms.

Explore scalar and vector fields in space, define temperature as a scalar field, and derive the gradient as a vector field that measures the maximum rate and direction of change.

Learn how outward flux through a closed surface defines divergence as the volume density of outward flux, derived from infinitesimal volumes and Gauss divergence theorem.

Explore line integrals of vector fields along paths, including closed-path circulation. Link circulation to curl via area vectors and the idea behind Stokes theorem.

Explore second derivatives using gradient, divergence, and curl rules, and introduce the laplacian operator del squared with identities like curl of gradient equals zero and curl of curl relations.

Explore line integrals of vector fields along paths from A to B, including open and closed paths. Learn path independence for gradient fields and the role of curl-free conditions.

Explore how to set up and compute volume integrals of scalar fields over cuboid and prism regions using dx dy dz, with examples like T = x y z^2.

Explore the fundamental theorems of gradient, divergence, and curl: gradient line integrals are path independent, Gauss's divergence theorem links flux to divergence, and Stokes theorem ties curl to boundary.

Explore orthogonal coordinate systems from Cartesian to spherical polar and cylindrical, derive transformation equations, and apply curl, divergence, and gradient in any coordinate system.

Delve into curvilinear coordinates, unit vectors, and scale factors; derive gradient, divergence, curl, and Laplacian, and learn transformations for spherical polar and cylindrical systems.

Explore how a Gaussian normalized to unit area approaches the Dirac delta as its width tends to zero, yielding a spike at x=0 with area one.

Shows how the vector field r / r^2 models a point charge at the origin, with divergence zero except at the origin, described by 3D Dirac delta and Gauss's theorem.

Explore how sources, charges at rest or in motion, create electric and magnetic fields, derive force on a test charge via Coulomb's law, and apply superposition to electrostatics.

Explores how identical charges at polygon vertices yield zero net force at the center by symmetry; also examines the z-axis field from two equidistant charges and the large-distance limit.

Apply the superposition principle and Coulomb's law to obtain the electric field from continuous charge distributions, using line, surface, and volume charge densities and vector calculus.

Derive the electric field above the midpoint of a finite uniformly charged line by integration, giving field along z-hat with magnitude (lambda 2L)/(4 pi epsilon0 z sqrt(z^2+L^2)).

Explore calculating the electric field from extended charges: rod, square loop, ring, and disk with uniform line or surface density at distance z above the center, using integration and symmetry.

Learn how Gauss's law connects the flux of the electric field through closed surfaces to the net charge enclosed, independent of surface shape, via divergence and superposition.

Apply Gauss's law to open and closed surfaces, using symmetry to relate flux through surface parts to the total flux for various charge configurations.

Apply Gauss law to spherical and cylindrical symmetry, determine electric field directions, compute magnitudes from charges enclosed, and analyze fields inside and outside spheres and cylinders.

Explore cylindrical and planar symmetry in electromagnetism, where electric fields align with the symmetry direction and depend only on distance from the axis or plane, via Gauss's law.

Show that the electrostatic field is curl-free and the gradient of a scalar potential defined by line integrals from a reference point, illustrating path independence and potential differences.

Learn to compute potentials for finite and infinite charge distributions, using path independence, reference points, and boundary conditions to relate inside and outside regions of shells, wires, and spheres.

Learn to compute electric fields and potentials from charge density through direct integration, Poisson’s equation, and Gauss’s law, while distinguishing finite and infinite distributions.

Explore boundary conditions in electromagnetism by showing how surface charge creates a discontinuity in the electric field's perpendicular component while parallel components remain continuous.

Explore work and energy in electrostatics: moving charges, external work, and the connection between potential, potential difference, and energy in point and continuous charge distributions.

Explore how a perfect conductor with unlimited free charge carriers rearranges charges to cancel internal fields, making the body equipotential and causing surface charges and induced charges.

Analyze how placing a charge inside a neutral conductor cavity induces inner and outer surface charges, yielding zero field inside and an external field equivalent to the net charge.

Delve into Laplace's equation for electrostatic potentials, covering boundary conditions, the Poisson equation, and the average-neighborhood property, with Earnshaw's theorem about stable equilibrium in charge-free regions.

Explore Laplace and Poisson equations, boundary value problems, and uniqueness theorems, plus the method of images for solving conductor problems.

Use the method of images to solve Poisson’s equation above a grounded plane, yielding V=(q/4πϵ0)(1/r_+−1/r_−) and the induced sigma(r)=−(q d)/(2π(r^2+d^2)^(3/2)) with total −q.

Extend the method of images to grounded planes, two semi-infinite conductors, and a grounded sphere, using image charges to satisfy boundary conditions and compute potentials, forces, and electrostatic energy.

Explore electrostatic capacitance and capacitor geometry, deriving capacitance as a purely geometrical quantity, linking potential differences, electric fields, and energy storage through harmonic potential theory.

Explore solving azimuth dosimetry potentials on a hollow sphere using Legendre polynomials to satisfy boundary conditions, ensuring continuity at the surface and finite interior and exterior solutions.

Explore solving electric fields and charge densities for line charges and conducting boundaries using Gauss's law and vector calculus, and analyze potentials and boundary conditions at infinity.

Derive and compare inside and outside potentials for electrostatic problems, apply Laplace's equation, and develop multipole expansions (monopole, dipole, quadrupole) using Legendre polynomials and generating functions.

Explore the multipole expansion of electrostatic potential, identifying when monopole, dipole, or quadrupole terms dominate at large distances and how dipole moment depends on origin.

Examine electrostatic energy and electric fields for complex charge configurations, including outside-inside regions, cavities in conductors, and dipoles on spherical surfaces, using potential and integration methods.

Explore how neutral atoms polarize in an electric field, inducing dipole moments and illustrating atomic polarization with crude atomic models and the relation between dipole moments and applied fields.

Explore magnetic fields from current-carrying wires and the Lorentz force on moving charges. See cyclotron and helix motions, and relate them to momentum and cyclotron frequency.

Explore the motion of a charged particle in perpendicular electric and magnetic fields, revealing a cycloid trajectory in the z-y plane, governed by v cross B and cyclotron frequency.

In a magnetic field, moving charges experience the Lorentz force q v × B, producing current I; the lecture defines line, surface, and volume current densities k and J.

Derive the continuity equation from charge conservation and relate surface current to the rate of change of enclosed charge. Introduce Biot-Savart law, steady current, and current enclosed by a path.

Analyze the force on a square loop in the y-z plane from a z-directed field using I×B, noting edge effects, and derive surface and volume current densities for rotating bodies.

Apply cross product and the superposition principle to magnetic fields from current loops, straight wires, and polygonal current distributions, with careful sign conventions and center-field calculations.

Explore Ampere's law with circular loops, relate line integrals to current enclosed, and derive the magnetic field of a long straight wire, highlighting surface choices and current continuity.

Explore Ampere's law and its applications in electromagnetism, as part of electromagnetism part-1. Apply amperian loops to analyze magnetic fields.

Apply Ampere's law to compute magnetic fields around an infinite straight wire and inside a solenoid, linking current to field in electromagnetism part-1.

Explore how current direction along z or phi affects the magnetic field direction, illustrated by solenoid configurations, to solidify the current-field relationship in electromagnetism.

Explore Ampere's law through Problem 5.14 and Problem 5.15, analyzing magnetic fields and the integral form in this 2024 discussion.

analyze how ampere law yields the magnetic field from coaxial solenoids with opposite currents, giving μ0 I (N2 − N1) inside the inner region and outer solenoid contributions elsewhere.

Explore the magnetic vector potential within electromagnetism, clarifying its role and applications in magnetic field analysis across the first part of the course.

Explore how magnetic forces do not work on moving charges, why batteries supply work instead, and how symmetry and line-integral reasoning explain fields inside and outside very long solenoids.

Explore how Maxwell's equations describe magnetic fields, the absence and hypothetical modification for magnetic monopoles, and the vector potential with gauge freedom.

derive magnetic dipole moment from a current loop and compute the dipole field via vector potential, explaining why magnetic monopoles do not exist and addressing non-uniform fields.

Explore magnetic fields in cylindrical geometries, using Ampere's law to analyze surface and volume current distributions inside and outside long cylinders, and relate magnetization and polarization to magnetic effects.

Explore how rotating charges generate magnetic dipole moments from current loops and spinning objects, and relate these moments to orbital and spin angular momentum.

Explore electromotive force and the line integral around a circuit, separating source and electric-field effects, with emphasis on battery regions, potential difference, and the static closed-loop condition.

Explore the physics of electromagnetic induction, analyzing line integrals, electric and magnetic fields, flux changes, and the generation of EMF in moving conductors.

Explore how changing magnetic flux induces emf in circuits, as moving conductors, sliding rails, and rotating magnets alter linked flux, with v × B guiding current directions.

Explore how changing magnetic flux induces current in a loop, governed by Faraday's law and Lenz's law, with surfaces and motion revealing the direction and magnitude of induction.

Examine the magnetic field of an infinitely long cylinder with uniform axial magnetization, deriving surface and volume bound currents and the resulting Ampere's law.

Explore mutual and self inductance by examining magnetic flux linkage between current loops, derive transformer principles, and apply surface and line integrals to compute flux and inductance.

Examine how changing currents produce magnetic flux and energy in magnetic fields, analyze inductance and mutual inductance, and apply Maxwell's equations to LC circuits.

Study a coaxial cylinder with surface currents, deriving the magnetic field between cylinders via Ampere's law and boundary conditions. Connect the analysis to Maxwell's equations, displacement current, and continuity equation.

Explore Maxwell's equations in vacuum and media, addressing divergence, boundary conditions, and polarization, and derive energy conservation via electromagnetic energy density and the Poynting vector.