ElectroMagnetic Theory Concepts & Solved Numericals

Q. 1 Find the gradient of these scalar fields : a) U = 4xz 2 + 3yz b) H = r 2 cosθ cosφ Ans: a) =>∇ U = 4z 2 a x + 3z a y + (8xz +3y) a z b) =>∇ H = 2r cosθ cosφ a r + r sinθ cosφ a θ + rcosθ sinφ a φ   Q.2 If V (x, y, z) = 3x 2 y –y 2 z 2 , find ∇ V and |∇ V| at the point (1, 2, -1). Ans: ∇ V = 6xy a x + (3x 2 – 2yz 2 ) a y + (-2y 2 z) a z At the point (1, 2, -1) ∇ V = 6(1)(2) a x + [3(12) – 2(2)(-1) 2 ] a y – 2(2) 2 (-1) a z =>∇ V = 12a x - a y + 8a z |∇ V| (1, 2, -1 ) = |12a x - a y + 8a z | = (209) 1/2 Q.3 Given φ = xy +yz +xz, find gradient φ at point (1, 2, 3) and the directional derivative of φ Ans: ∇ φ = (y + z)a x + (x + z)a y + (y + z)a z At point (1, 2, 3) ∇ φ = 5a x + 4a y + 3a z The directional derivative is given as: dφ/dl = ∇ φ . a l = (5, 4, 3) . [(3, 4, 4) – (1, 2, 3)] / 3 = [(5, 4, 3) . (2, 2, 1)] / 3 = 7 Q.4 Find V (x, y, z) if grad V = (y 2 – 2xyz 3 )a x + (3 ...

- The gradient of a scalar field provides a vector field that states how the scalar value is changing throughout space – a change that has both a magnitude and direction. - A gradient is applied to a scalar quantity that is a function of a 3D vector field: position . The gradient measures the direction in which the scalar quantity changes the most, as well as the rate of change with respect to position. - The physical meaning of the gradient of a scalar is that it represents the steepness of the slope or line. For example, height is a scalar quantity; gradient of the height would be a vector pointing upwards. The length of the vector is proportional to the steepness of the slope. - A derivative is required that tells us how fast the function varies , if we move a little distance. - Consider a scalar function T which is a function of space coordinates x, y and z. - The projection or the component of ∇ T in the direction of a unit vector a l is ∇ T . a l and...