ElectroMagnetic Theory Concepts & Solved Numericals

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...