ElectroMagnetic Theory Concepts & Solved Numericals

Geographers specify a location on the Earth’s surface using three scalar values: longitude, latitude, and altitude. Both longitude and latitude are angular measures , while altitude is a measure of distance . Latitude, longitude, and altitude are similar to Spherical Co-ordinates. Spherical coordinates consist of one scalar value (r), with units of distance, while the other two scalar values (θ, φ) have angular units (degrees or radians). A Vector in Spherical System is represented as (A r A θ , A φ ) or A = A r a r + A θ a θ + A φ a φ Where a r , a θ and a φ are the unit vectors in r, θ and φ direction respectively. The physical significance of each parameter of spherical coordinates: - The value r expresses the distance of the point from origin (i.e. similar to altitude). It is the radius of the sphere. - The angle θ is the angle formed with the z- axis (i.e. similar to latitude). It is also called the co-latitude angle. It is measured clockwise.

Differential Length(dl):   In General the differential length is given as dl = dρ a ρ + ρdφ a φ + dz a z Differential Length for a surface is given as: - dl = dρ a ρ + ρdφ a φ ---(For ρ-φ Plane or Z constant Plane) - dl = ρdΦ a φ + dz a z ---(For φ-z Plane or ρ Constant Plane) - dl = dρ a ρ + dz a z ---(For ρ-z Plane or φ Constant Plane) Differential length for a line parallel to ρ, φ and z axis are respectively given as: - dl = dρ a ρ ---(For a line parallel to ρ axis) - dl = ρdφ a φ ---(For a line parallel to φ axis) ∫ dl = ∫ o 2π ρdφ = ρ( 2π - 1) = 2πρ This resembles the circumference of a circle. Hence if φ varies with ρ and z constant, then the length is the circumference of the circle. dl = dza z ---(For a line parallel to z axis) Differential Surface (ds): - ds = ρdρ dφ a z   This surface describes a circular disc. Always remember- To define a circular disk we need two parameter one distance measure and one angular measure. An

A Vector in Cylindrical system is represented as (A ρ , A φ , A z ) or A = A ρ a ρ + A φ a φ + A z a z Where a ρ , a φ and a z are the unit vectors in ρ, φ and z direction respectively. The physical significance of each parameter of cylindrical coordinates:      - The value ρ indicates the distance of the point from the z-axis. It is the radius of the cylinder .      - The value φ, also called the azimuthal angle , indicates the rotation angle around the z-axis. It is basically measured from the x axis in the x-y plane. It is measured anti-clockwise.      - The value z indicates the distance of the point from z-axis. It is the same as in the Cartesian system. In short, it is the height of the cylinder. Range of the variables: It defines the minimum and the maximum value that ρ, φ and z can have in Cartesian system.  0 ≤ ρ ≤ ∞  0 ≤ φ ≤ 2π - ∞ ≤ z ≤ ∞   Figure shows Point P and Unit vectors in Cylindrical Co-ordinate System. Cylindrical System - Unit v

A Vector in Cartesian system is represented as (A x , A y , A z ) Or A = A x a x + A y a y + A z a z Where a x , a y and a z are the unit vectors in x, y, z direction respectively. Range of the variables: It defines the minimum and the maximum value that x, y and z can have in Cartesian system. -∞ ≤ x,y,z ≤ ∞ Differential Displacement / Differential Length (dl): It is given as dl = dxa x + dya y + dza z Differential length for a surface is given as: dl = dxa x + dya y --- ( For XY Plane or Z Constant Plane). dl = dya y + dza z , ---( For YZ Plane or X Constant Plane). dl = dxa x + dza z ---( For XZ Plane or Y Constant Plane). Differential length for a line parallel to x, y and z axis are respectively given as: dl = dxa x ---( For a line parallel to x-axis). dl = dya y ---( For a line Parallel to y-axis). dl = dza z , ---( For a line parallel to z-axis). If there is a wire of length L in z-axis, then the differential length is given as

In order to describe the spatial variations of the quantities, appropriate coordinate system is required. - A point or vector can be represented in a curvilinear coordinate system that may be orthogonal or non-orthogonal. - An orthogonal system is one in which the coordinates are mutually perpendicular to each other. - The different types of orthogonal co-ordinate system available are: Cartesian or Rectangular. Circular Cylindrical. Spherical. Elliptical Cylindrical. Hyperbolic Cylindrical. Parabolic Cylindrical . The choice depends on the geometry of the application. - The frequently used and hence discussed herein are Rectangular Co-ordinate system. (Example: Cube, Cuboid) Cylindrical Co-ordinate system. (Example : Cylinder) Spherical Co-ordinate system. (Example : Sphere) - A set of 3 scalar values that define position and a set of unit vectors that define direction form a co-ordinate system. - The 3 scalar values used to define position are call