ElectroMagnetic Theory Concepts & Solved Numericals

INDEX Vector analysis is a mathematical tool with which electromagnetic (EM) concepts are most conveniently expressed. It's important to learn its rules and techniques first applying it. VECTOR ALGEBRA: - Scalars and Vectors. - Unit Vectors. - Position and Distance Vectors. - Vector Multiplication. - Components Of a Vector. - Numericals / Solved Examples. COORDINATE SYSTEMS & TRANSFORMATION: In general, the physical quantities in ElectroMagnetics are functions of space and time. In order to describe the spatial variations of the quantities, its important to define all points uniquely in space in a suitable manner. This requires using an appropriate coordinate system. Hence its very important to understand the coordinate system first.  - Introduction To Coordinate System. - Cartesian Coordinate System / Rectangular Coordinate System (x, y, z). - Differential Analysis Of Cartesian Coordinate System. - Circular Cylindrical Coordinate System (ρ, φ, z)...

Q.1) Given vectors A = a x + 3a z and B = 5a x + 2a y -6a z , determine a) |A+B| b) 5A – B c) Component of A along a y . d) Unit vector parallel to 3A + B. Ans: a) A + B = a x +3a z +5a x +2a y - 6a z = 6a x + 2a y – 3a z . |A + B| = (6 2 + 2 2 + 3 2 ) 1/2 = 7. b) 5A – B = 5(a x + 3a z ) – 5a x – 2a y + 6a z = -2a y + 21a z . c) Component of A along a x , a y and a z are 1 , 0 and 3 respectively. d) 3A + B = 3a x + 9a z + 5a x + 2a y -6a z = 8a x + 2a y +3a z a 3A + B = 3A + B/ |3A + B| = (8a x + 2a y + 3a z )/ (77) 1/2 Q.2) Given Points P (1, -3, 5), Q (2, 4, 6) and R (0, 3, 8) find a) Position vectors of P and R. b) Distance vector r QR . c) Distance between Q and R . Ans: a) r p = a x – 3a y + 5a z . r R = 3a y +8a z . b) r QR = r R -r Q = 3a y +8a z – (2a x + 4a y + 6a z ) = - 2a x – a y + 2a z . c) |r QR | = (2 2 + 1 2 + 2 2 ) 1/2 = 3. Q.3) Find the unit vector along the...

Vector Multiplication: When two vectors are multiplied the result is either a scalar or a vector depending on how they are multiplied. The two important types of vector multiplication are: Dot Product/Scalar Product (A.B) Cross product (A x B) DOT PRODUCT (A . B): - Dot product of two vectors A and B is given as: A . B = |A| |B| cosθ AB Where θ AB is the angle formed between A and B. Also θ ranges from 0 to π i.e. 0 ≤ θ AB ≤ π - The result of A.B is a scalar , hence dot product is also known as Scalar Product. - If A = (A x , A y , A z ) and B = (B x , B y , B z ) then A.B = A x B x + A y B y + A z B z - If A.B = |A| |B|, then obviously cosθ AB =1 which means θ AB = 0 o This shows that A and B are in the same direction or we can also say that A and B are parallel to each other. - If A.B = - |A| |B|, then obviously cosθ AB = -1 which means θ AB = 180 o . This shows that A and B are in the opposite direction or we can also say that A and B ar...

Most of the physical quantities are either scalar or vector quantities. SCALAR QUANTITY: - Scalar is a number that defines magnitude. Hence a scalar quantity is defined as a quantity that has magnitude only. - A scalar quantity does not point to any direction i.e. a scalar quantity has no directional component. For example when we say, the temperature of the room is 30 o C, we don’t specify the direction. - Hence examples of scalar quantities are mass, temperature, volume, speed etc. - A scalar quantity is represented simply by a letter – A, B, T, V, S. VECTOR QUANTITY: - A Vector has both a magnitude and a direction. Hence a vector quantity is a quantity that has both magnitude and direction. - Examples of vector quantities are force, displacement, velocity, etc. - A vector quantity is represented by a letter with an arrow over it. UNIT VECTORS (a A ): - When a simple vector is divided by its own magnitude, a new vector is created known as the unit vect...