Tips & Tricks to get the concept of Electromagnetic Field Theory Significance : 1

Dear aspirants of GATE/ESE 2018, we are going to take a New Initiative which will help to boost your preparation.In this article, we are going to provide only the "Revision & Concepts of Electromagnetic Field Theory", in turn, you will get an enhancement in your preparation for GATE/ESE 2O18. It's a quick & smart revision for this subject we wish you will get the Fruitful Result after going through the concept in this article.

In future or in upcoming days we will start a live session on concepts to the same & other subjects.In this article, we are going to provide the significance of Vector Analysis & Significance of the Different Types Operator which generally occurs in in Electromagnetic Field Theory.Here in this article, we will go in detailed information about the subject but definitely provide the ways in which direction you should proceed to prepare Electromagnetic Field theory.

1. Rectangular Cartesian System

• This is the most common and often preferred coordinate system is defined by the intersection of three mutually perpendicular planes.
• Once an origin is selected with coordinate (0, 0, 0), any other point in the plane is found by specifying its x-directed, y-directed, and z-directed distances from this origin.

• Coordinate directions are represented by unit vectors i_x, i_y, and i_z, each of which has a unit length and points in the direction along one of the coordinate axes.

Circular Cylindrical Coordinates

• The cylindrical coordinate system is convenient to use when there is a line of symmetry that is defined as the z-axis.
• If any point in space is defined by the intersection of the three perpendicular surfaces of a circular cylinder of radius r, a plane at constant z, and a plane at constant angle Φ from the x-axis.
• The unit vectors i_r, i_Φ, and i_z are perpendicular to each of these surfaces.
• The direction of i_z, is independent of position, but unlike the rectangular unit vectors the direction of i_r and i_Φ, change with the angle Φ.

For instance, when Φ = 0 then i_r=i_x, and i_y=i_Φ,, while if Φ=π/2, then ⇒ i_r=i_y, and i_Φ = -i_x.

Note: By convention, the triplet (r, Φ, z) must form a righthanded coordinate system so that curling or say rotating the fingers of the right hand from i_r, to i_Φ puts the thumb in the

z-direction.

Differential Size Cylindrical Volume

A section of the differential size cylindrical volume is formed when one moves from a point at coordinate (r, Φ, z) by an incremental distance d_r, rd_Φ, and d_z in each of the three coordinate directions. 

Differential Lengths, Surface Area, and Volume Elements for Each Geometry

Spherical Coordinates System

• A spherical coordinate system is useful when there is a point of symmetry that is taken as the origin.
• Spherical coordinate (r, θ, Φ ) is obtained by the intersection of a sphere with radius r, a plane at a constant angle Φ from the x-axis as defined for the cylindrical coordinate
  system, and a cone at angle θ from the z-axis. 
• The Component i_r, i_θ, and i_Φ are perpendicular to each of these surfaces and change direction from point to point.
• Here also the triplet (r, θ, Φ ) must form a right-handed set of coordinates.
• Here the incremental displacements d_r, rd_θ, rsinθ.d_Φ from the coordinate (r, θ, Φ ) now depends on the angle θ and the radial position r.

VECTOR ALGEBRA

Scalars and Vectors

A scalar quantity is a number completely determined by its magnitudes, such as temperature, mass, and charge, the last being especially important in our future study

 A Vectors, such as velocity and force, must also have their direction specified and in this text are printed in boldface type. They are completely described by their components along three coordinate directions. 

• A vector is represented by a directed line segment in the direction of the vector with its length proportional to its magnitude.

The vector is then A = A_xi_x+A_yi_y+A_zi_z

& its Magnitude |A|= [A_x²+A_y²+A_z²]^1/2

• Note that each of the components in  (A_x, A_y, and A_z) are themselves scalars.
• The direction of each of the components is given by the unit vectors.
• We could describe a vector in any of the coordinate systems replacing the subscripts (x, y, z) by (r,Φ, z) or (r,θ,Φ) then.

Multiplication of a Vector by a Scalar

If the Original Vector is The vector is then A = A_xi_x+A_yi_y+A_zi_z then after multiplying a scaler Quantity "a" then 

aA = a(A_xi_x)+a(A_yi_y)+a(A_zi_z)

Addition and Subtraction 

The sum of two vectors is obtained by adding their components while their difference is obtained by subtracting their components.

If the vector B is  B = B_xi_x+B_yi_y+B_zi_z

If B  added or subtracted then

C = A+B =  (A_x±B_x)i_x+ (A_y±B_y)i_y+ (A_z±B_z)i_z

• Geometrically, the vector sum is obtained from the diagonal of the resulting parallelogram formed from A and B as shown in Figure.
• While  The difference is found by first drawing -B and then finding the diagonal of the parallelogram formed from the sum of A and -B. 
• The sum of the two vectors is equivalently found by placing the tail of a vector at the head of the other.
• Subtraction is the same as addition of the negative of a vector.

Addition of Two Vector

Subtraction of Two Vector

The Dot (Scalar) Product

⇒ The dot product between two vectors results in a scalar and is defined as 

A^.B=A^.B cosθ

Here The term "Acosθ" is the component or say Vector Projection of the vector A in the direction of B.

Application of Dot Product

One application of the dot product arises in computing the incremental work dW necessary to move an object a differential vector distance dl by a force F.

dW = F^.dL

product between a unit vector and itself is unity

i_x^.i_x = 1 while i_x^.i_y =0 similarly

i_y^.i_y = 1 while i_y^.i_z =0

i_z^.i_z = 1 while i_z^.i_x =0

Then the dot product can also be written as

A^.B = (A_xi_x+A_yi_y+A_zi_z)^.(B_xi_x+B_yi_y+B_zi_z)

Note: we see that the dot product does not depend on the order of the vectors.

i.e; A^.B = B^.A

The angle between vectors is The Cross (Vector) Product

⇒ The cross product between two vectors "A x B" is defined as a vector perpendicular to both A and B, which is in the direction of the thumb when using the right-hand rule of curling the fingers of the right hand from A to B as shown in Figure.

The magnitude of the cross product is

|AxB|=AB Sinθ

where θ is the enclosed angle between A and B. 

• Geometrically the above equation gives the area of the parallelogram formed with A and B as adjacent sides. Interchanging the order of A and B reverses the sign of the cross product.

AxB = - BxA

For rectangular unit vectors, we have These relations allow us to simply define a right-handed coordinate system as one where

i_x x i_y = i_z

Similarly, for cylindrical and spherical coordinates, righthanded coordinate systems haveThe relation of cross product allows us to write 

Which can be compactly expressed as the determinantal expansion

Note: If we think of xyz as a three-day week where the last day z is followed by the first-day x, the days progress as

• Where the three possible positive permutations are underlined. Such permutations of xyz in the subscripts in the determinant form have positive coefficients while the odd permutations, where xyz do not follow sequentially like xzy, yxz, zyx which have negative coefficients in the cross product. 

 Co-Ordinate System & Their Interrelationship

Geometric relations between coordinates and unit vectors for Cartesian, Cylindrical, and spherical coordinate systems. 

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