Introduction to Electromagnetics - Rahsoft RAHAE101

Explore vector basics, distinguishing magnitude and direction with unit vectors and components along X, Y, and Z. Apply these ideas to the electric field intensity and compute magnitude as sqrt(X^2+Y^2+Z^2).

Explore vector addition and subtraction using component-wise methods, parallelogram law, and vector algebra rules to compute the resultant and position vectors.

Explore the distance vector as the displacement between points, express it as r_A minus r_B, and practice calculating magnitude, direction, components, and unit vectors.

Practice calculating vector components and magnitudes, derive the unit vector and projections for A and B, and compute position vectors and distances such as B to Q.

Calculate distances and distance vectors in three-dimensional space using modulus, position vectors, and unit vectors; determine the distance between P and Q and vectors parallel to PKU with specified magnitudes.

Solve a quiz of vector problems in electromagnetics, finding unit vectors, directions, magnitudes, and midpoints, including origin-to-point and vector-field directions.

Compute unit vectors and magnitudes from given vectors, determine directions, and solve for vector components using subtraction and algebra to obtain the final vector and its direction.

Navigate a quiz solution by solving a set of equations with X and Y, computing RMN and RMV, and deriving a surface and line relationships through algebraic manipulation.

Solve quiz problems in electromagnetics using vector notation, find coordinates for A, B, and C, and compute magnitudes, unit vectors, and the midpoint and line direction from the origin.

Explore dot and cross products: the dot product yields a scalar projection along a unit vector; the cross product yields a vector whose magnitude equals the parallelogram area.

Explore the cross product of unit vectors, applying the right-hand rule to determine direction, and examine its noncommutative, nonassociative, and distributive properties, including sign changes for clockwise versus anticlockwise rotation.

Understand the scalar triple product, represented as a determinant and as the dot product A · (B × C), and see how cross and dot products form this relation.

Explore the vector triple product by representing cross products with a determinant, using i, j, k components, and solving to verify the expansion.

practice finding the angle between vectors a and b using dot and cross products, with a = 3x+4y+z and b = 2y−5z, via magnitudes and trigonometric relations.

Compute vector operations in electromagnetics, including dot and cross products, determine magnitudes and the angle between vectors, and find the parallelogram area and perpendicular unit vectors.

Compute the angle between vectors via the cross product, determine the cross product magnitude, obtain a unit vector perpendicular to u and r, and project B along Q.

Explore vector operations in electromagnetics, including dot product, cross product, and the law of cosines, and apply triangle area, sine rule, and distance calculations through a quiz discussion.

Solve quiz problems on vector operations in electromagnetics, using cross products and determinants to find vectors orthogonal to a plane and test whether vectors lie in the same plane.

Tackle multiple choice questions on vectors and scalars in this electromagnetics primer, analyzing magnitude versus direction, right-handed coordinate systems, cross products, and vector projections to solve consistency and validity checks.

Explore coordinate systems and transformation, including right-handed Cartesian, cylindrical, and spherical systems, and develop orthogonal curvilinear coordinates, enabling gradient, divergence, and Laplacian calculations.

Explore the Cartesian coordinate system, with x, y, and z axes, and locate points by plane intersections and coordinates like (3,4,5).

Explore the cylindrical coordinate system for solving electromagnetic problems, defined by radial distance r, azimuthal angle phi, and height z, with orthogonal surfaces and unit vectors er, e_phi, ez.

Explore the range of cylindrical coordinates, where r ranges from zero to infinity, and understand unit vectors and their orthogonality, magnitude, dot product, and cross product relationships.

Explore the relationship between cartesian and cylindrical coordinates, showing r equals the root of x squared plus y squared and phi equals arctan(y/x), with cartesian and cylindrical coordinate transformations.

Explore the relationship between Cartesian and cylindrical unit vectors, and how components transform via angle-based expressions using cosine and sine to relate x and y components.

Learn the relationship between Cartesian and cylindrical vectors, derive forward and inverse transformations using matrix form, and express components in Cartesian and cylindrical coordinates.

Explore the spherical coordinate system, including radius and the cone and plane constraints that locate points on a sphere, and compare with cylindrical coordinates.

Explore the spherical coordinate system defined by a sphere, a cone, and a plane of constant angle, with unit vectors perpendicular to these surfaces along increasing coordinates.

Explore the spherical coordinate unit vectors and vector representation. See how Cartesian, cylindrical, and spherical coordinates relate and transform between x, y, z and r, theta, phi.

Demonstrates transforming coordinates between Cartesian and cylindrical systems, and shows projecting a point onto the xy plane to relate axis directions and angles.

Transform points between Cartesian and cylindrical coordinates and map cylindrical coordinates to Cartesian coordinates, highlighting the x squared plus y squared relation.

Explore the transformation between cylindrical and Cartesian coordinates, converting points between systems using the table, and extend to transforming vectors between these coordinate systems.

Learn how to transform a vector from Cartesian to cylindrical coordinates, express components along cylindrical unit vectors, and relate magnitude to angle between axes.

Develop the technique to transform vectors from Cartesian to cylindrical coordinates by evaluating dot products of unit vectors, angles, and projections to establish component relationships.

Learn how to transform cartesian coordinates to cylindrical coordinates and express the vector transformation in matrix form, using the given equations.

Learn how to transform vectors from cylindrical to Cartesian coordinates, using coordinate relationships and matrix form to convert vector representations between systems.

Learn how to transform points and vectors between Cartesian and spherical coordinate systems, and perform forward and inverse conversions for Cartesian components to spherical coordinates and back.

Transform the vector from cartesian coordinates to spherical coordinates, deriving the relationships between x, y, z and the spherical components, and using unit vectors to project onto the spherical axes.

Transform a vector from spherical to cartesian coordinates, deriving the cartesian components from spherical variables and applying the conversion to obtain x, y, and z expressions.

Explore transforming a vector from spherical to Cartesian coordinates in electromagnetics, using explicit formulas and component relationships, and apply a simplified table to solve numerical problems.

Transform coordinates between Cartesian, cylindrical, and spherical systems using numerical examples that illustrate conversion formulas, such as rho equals sqrt(x^2+y^2) and phi equals arctan(y/x).

Learn how to transform a point from Cartesian coordinates to cylindrical coordinates, computing r and the angle, and convert back to verify the relation between x, y, z.

Convert a Cartesian point to cylindrical and vertical coordinates, compute the radial distance r and the associated angles, and verify the angle values in degrees through step-by-step calculations.

Compute the cartesian coordinates from cylindrical points and solve the distance between points B and Q, using coordinate transformations, distance formulas, and unit vector components.

Transform cartesian vectors to cylindrical coordinates by computing r, phi, and z components and translating unit vectors with x, y, z relations.

Derives how a given vector converts to cartesian components, computes x and y projections, and expresses the vector in cartesian form using unit vectors.

Solve a numerical electromagnetics problem by resolving a vector into its x, y, z components and converting the result to spherical components.

The lecture demonstrates converting Cartesian coordinates to spherical coordinates using the x^2 + y^2 + z^2 formula, computing r ≈ 5.38 and angles around 56.3 degrees using the example.

Continue solving a numerical electromagnetics problem by applying trigonometric calculations in degrees, obtaining sine values and a final solution toward unit iv.

Explore the transformation of a vector from Cartesian coordinates to cylindrical and spherical polar coordinates, deriving components and expressions using the metric form.

Explore the transformation from the Cartesian coordinate system to a local coordinate system and simplify the resulting expressions in numerical electromagnetics.

Apply numerical methods to convert spherical coordinates to Cartesian coordinates and compute a 3d point using vector components and trigonometric relations.

Continue from problem 13 by solving the system, substituting values into equations five and six to obtain X = -2 and Y = 1.

solve numerical 14 by transforming between coordinate systems, including vertical and cylindrical frames, deriving and calibrating point coordinates step by step to obtain the solution.

Explore solving problems on transforming between coordinate systems—Cartesian, cylindrical, and spherical—expressing given equations in each form and locating points across systems.

Solve queries 1 and 2 by transforming between Cartesian and cylindrical coordinates, derive the relationships among x, y, and r, and compare the governing equations.

Explore solving coordinate transformations in electromagnetics, translating between Cartesian, spherical, and cylindrical systems, and deriving unit vectors and coordinate components for queries 3 and 4.

Transform Cartesian coordinates to cylindrical and spherical forms, apply the relations to solve queries 5, 6, and 7, and determine the vector components at (1,1,1).

Solve queries 8 and 9 by deriving Cartesian, cylindrical, and spherical coordinate relations, converting variables, and expressing field components in different coordinate systems.

Solve queries 10 and 11 by converting a point from Cartesian to cylindrical and spherical coordinates, then transform back to Cartesian to illustrate coordinate transformations.

Explore differential length, area, and volume in Cartesian and local coordinate systems, deriving the differential length vector and its magnitude, and preview differential surface area calculations.

Explore differential surface area and differential volume in Cartesian coordinates. Treat area as a vector normal to the surface, and multiply three differential lengths to obtain volume.

Explore differential length, area, and volume in the cylindrical coordinate system, including how incremental changes define line, surface, and volume elements and their normal orientations.

Explore the spherical coordinate system and derive differential length, area, and volume elements. Understand how radius and angles define surface and volume in electromagnetics.

Explore solved problems on differential length, volume, and area using cylindrical and spherical coordinates, with double and triple integrations to compute surfaces and solid regions.

Explore line integrals of vector fields along a curve from a to b, using F·dr in Cartesian form, and note that closed-path line integrals vanish.

Explore surface integrals by computing the flux of vector A through a surface using the outward unit normal. Learn the double integral notation for surface flux.

Understand how a closed surface in space encloses a volume and apply the triple integral over that volume, with practice problems to reinforce the concept.

Explore how to compute line, surface, and volume integrals in vector fields to determine work, using F, unit vectors, and limit-based integration techniques.

Compute the vector field flux through the cube by performing face-by-face surface integrals over its six faces and summing the results to obtain 3/2.

Compute the line integral around the path by evaluating four segment contributions, each using unit vectors and x,y coordinates, and obtain a total circulation of minus one sixth.

This lecture presents solved problem 4 from introduction to electromagnetics, outlining a region bounded by x=0 and y=0 and evaluating an integral with x and y unit vectors, yielding 32/15.

Present the solution to problem five in introduction to electromagnetics, solving line and surface integrals for vector field A = 4x i + 9y j across parts one through three.

Solve problem six by evaluating surface integrals over unit cube using r = x i + y j + z k, summing faces to yield total flux of three.

Solve problems 7 and 8 demonstrate using divergence theorem and Gauss's law to evaluate flux through a sphere and determine enclosed charge and electron count from a spherical density distribution.

Learn how to differentiate functions of multiple variables using partial derivatives, compute the total differential, and apply the del operator in Cartesian and cylindrical coordinates.

Understand how the gradient of a scalar field points in the direction of steepest ascent, equals x, y, z partial derivatives, and has magnitude as the maximum rate of change.

Explore the gradient of V in different coordinate systems, including cylindrical and vertical coordinates, and discuss the physical significance of describing the gradient.

Explain the physical significance of the gradient: it has magnitude and direction and points to the maximum rate of change of a scalar field like the potential V.

Examine the divergence of a vector field as the net outward flux per unit volume, and its expression via del dot A and the surface flux through a closed surface.

Learn how divergence of a vector field measures net outward flux per unit volume as a region shrinks to a point, with incompressible water (zero divergence) and gas flow (nonzero).

Explore the divergence of a vector in Cartesian and other coordinate systems. See explicit representations and how the divergence representation differs across coordinate systems.

Explore how curl links circulation around a loop to the rotation of a vector field, using circular motion, axis of rotation, and the normal to the oriented area.

Analyze curl and its physical significance in electromagnetics, illustrating how vector fields circulate around points and relate to boundary behavior.

Derive the curl in three-dimensional space by linking surface integrals to vector calculus concepts, and evaluate the curl at points with coordinates x, y, z.

explain the physical significance of curl across different coordinate systems, and show how rotation relates to velocity fields, including examples in cylindrical and spherical coordinates to solve problems.

Explore gradient calculations through solved problems, compute gradient components, substitute points, and apply e = -∇v to find the y-component of the electric field.

Walk through solved example 3 in introduction to electromagnetics, derive and simplify a vector field expression built from x^2+y^2 terms, compute the gradient, and introduce divergence problems for practice.

Explore solved examples 4 and 5, deriving gradient and divergence of fields with partial derivatives, and applying unit vectors to simplify expressions.

Solve problems six and seven by applying divergence in cylindrical coordinates, performing the required differentiations and substitutions to illustrate the method.

Solve examples 8 and 9 by applying divergence in spherical and cylindrical coordinates, substituting values and differentiating, and evaluating at a given point to obtain -1.

Solve examples 10 and 11 by deriving divergence expressions in cylindrical and spherical coordinates, differentiating vector fields with respect to r, theta, and phi, and applying to position vector r.

Learn Stokes theorem by equating the line integral around a closed curve to the surface integral of a vector field over the surface bounded by the curve.

Demonstrates Stokes theorem by solving a line integral and the corresponding surface integral in cylindrical coordinates, confirming equality and highlighting differential surfaces and coordinate expressions.

Apply Stoke's theorem to relate a surface integral to a line integral over a quarter-circle boundary, compute using parametrization, and verify the equality.

Apply Stoke's theorem by relating the surface integral of curl to the line integral around a square boundary in the x–y plane, considering boundary orientation.

This lecture continues the Stokes theorem quiz, solving problems that connect line integrals to surface integrals, compute curl, and recognize when Stokes theorem applies.

Apply Green's theorem to relate divergence-based volume integrals to surface integrals and boundary terms, showcasing the equivalence between volume and surface expressions in electromagnetics.

Helmholtz's theorem states that any vector field in a region with boundary values splits into a divergence-free part and a curl-free part, via a vector potential and a scalar potential.

Explore the Laplacian of a scalar, the divergence of its gradient, and its form in Cartesian, cylindrical, and spherical coordinates, noting that it is harmonic when the Laplacian equals zero.

Demonstrate numerical solutions to the laplacian of a scalar function, using cylindrical coordinates to derive and simplify the laplacian expression and verify zero conditions through examples.

Explore numerical approaches to the Laplacian of a scalar field, simplify related equations, and derive solutions through step-by-step examples.

Study electrostatic field basics with stationary charges and Coulomb's law. Examine electric force between charges, proportional to the product and inversely to the square of distance.

Explore Coulomb's law: compute forces between point charges using vector form and unit vectors, apply superposition, and see how media like air or vacuum and epsilon affect the force.

Tackles a numerical problem on Coulomb's law: calculating the electrostatic force between two charges in a dielectric medium using 1/(4 pi epsilon0) and the inverse square distance.

This lecture demonstrates a Coulomb's law numerical problem using the vector form F = k q1 q2 (r_A − r_B)/|r_A − r_B|^3, with position vectors A and B.

Apply coulomb's law to a three-charge equilateral-triangle setup to compute the magnitudes FCA and FCB, then determine the resultant force on q3 with vector addition.

Explore numerical applications of Coulomb's law with suspended charges, deriving forces, energy, and equilibrium from separation x and epsilon.

Explore electric field intensity, defined as the force on a positive test charge in a region around charges, with examples and the vector form for point and multiple charges.

Compute the electric field intensity at a point due to a system of charges by summing each charge's contribution kq/r^2, where r is the distance to the point.

Explore an example computing the electric field intensity at a point due to two point charges Q1 and Q2, using vectors and epsilon0. The solution shows components and magnitude.

Explore how line, surface, and volume charge densities determine electric fields, derive expressions for line, surface, and volume charge distributions, and learn the corresponding integral formulas for E.

Explore numerical methods in electromagnetics to compute the electric field from a uniformly charged circular disk using surface integration, symmetry, and epsilon naught.

Compute the electric field of a uniformly charged circular disk in the xy-plane by integrating surface elements, extracting axial and radial components, and noting infinite-plane limits.

Explore numerical methods for calculating the electric field from a line charge distribution along an infinite axis, using the 1/r^2 law and integration to obtain field components.

Analyze numerical charge distributions for line charges, using symmetry and cylindrical coordinates to compute electric fields, with finite and infinite lines and a -4 to 4 segment.

Determine the electric field due to an infinite plane with a given surface charge density on the side containing the origin, using the unit normal to express the field components.

Derives the electric field intensity from a uniformly charged line segment between x=0 and x=3 using integration, computing the differential field dE and evaluating the resulting expression.

The lecture covers numerically analyzing charge distributions on planes with uniform surface charge, computing the electric field at specified points, and solving the resulting equations to find field values.

Explore the electric field of an infinitely long line charge using Cartesian coordinates, deriving the field by integrating contributions from infinitesimal elements and exploiting symmetry to cancel horizontal components.

Derive the electric field of an infinitely long line charge in cylindrical coordinates, showing only the radial component contributes and the field varies with distance from the wire.

Explore the electric field from a finite line of charge in Cartesian coordinates, showing horizontal components cancel and integrating along the wire yields vertical field related to linear charge density.

Learn how to compute the electric field from a finite line of charge using cylindrical coordinates, applying symmetry to simplify the field and derive the integral expression.

Derive the electric field intensity of a sheet of surface charge by dividing it into strips and integrating their contributions to obtain E = sigma/(2 epsilon0) outward from the sheet.

Explore how the electric field intensity from a volume charge within a sphere is derived by integrating differential charge elements over the volume.

Explore electric flux density (displacement flux) D and its relation D = ε E in free space and media, with a point-charge perspective and the role of epsilon.

Explore electric flux density for an isolated point charge, deriving surface-integral flux and the 1/(4π ε0) relation via Gauss's law.

Sum the charges over the surface to compute the total flux through a closed surface, then apply Gauss's law with epsilon naught to determine the electric flux density.

this lecture presents a numerical approach to electric flux density on a closed surface, deriving D from epsilon and area, and evaluating surface integrals across specified bounds.