Question

Question 2

(a) Given that A ~ = (x + 2y + 4z)ˆ

i + (2x - 3y- z)ˆj +(4x-y + 2z)kˆ

(i). Show that the vector field A ~ is irrotational

(ii). Find the scalar potential φ such that A~ = ∇φ, if φ(0, 0, 0) = 1

Answer 1

1)

$\nabla ×\vec A=\vec 0,$

$\nabla×\vec A=(\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z})\vec i+(\frac{\partial A_x}{\partial z}-\frac{\partial A_z}{\partial x})\vec j+(\frac{\partial A_y}{\partial x}-\frac{\partial A_x}{\partial y})\vec k=(-1+1)\vec i+(4-4)\vec j+(2-2)\vec k=\vec 0,$

2)

$\nabla \varphi=\frac{\partial A}{\partial x}\vec i+\frac{\partial A}{\partial y}\vec j+\frac{\partial A}{\partial z}\vec k=\vec i-3\vec j+2\vec k.$