Topic: Electromagnetics
For a conservative vector field F below, which of the following is the scalar potential? $F=(y^2\cos x+az^3)\vec{i}+(2y\sin x-4)\vec{j}+(3xz^2+2)\vec{k}$ (a is an integer)
- A. Cannot be found
- B. $xz^3+y^2\sin x-4y+2z$
- C. $2 y \cos x-y^2\sin x-2y+4z$
- D. $2y\sin x-z^2\cos x-2y+4z$
Answer: B. $xz^3+y^2\sin x-4y+2z$
Explanation: For a field to be conservative, its curl must be zero ($\nabla \times \vec{F} = 0$), which necessitates $a=1$. The scalar potential $V$ satisfies $\vec{F} = \nabla V$. Integrating the components: $\int (y^2\cos x + z^3)dx = y^2\sin x + xz^3 + f(y,z)$. Differentiating with respect to y gives $2y\sin x + \frac{\partial f}{\partial y}$, which must equal the j-component $(2y\sin x - 4)$. Thus, $f(y,z) = -4y + g(z)$. Differentiating with respect to z gives $3xz^2 + g'(z)$, which must equal the k-component $(3xz^2 + 2)$. Thus, $g(z) = 2z$. The final scalar potential is $y^2\sin x + xz^3 - 4y + 2z$.