Consider two electrostatic fields in a region of free space given by the vector functions $\vec{E}_1

Question

Consider two electrostatic fields in a region of free space given by the vector functions $\vec{E}_1 = 2axy\hat{i} + a(x^2 - y^2)\hat{j}$ and $\vec{E}_2 = ay\hat{i} + (ax + bz)\hat{j} + by\hat{k}$, where $a$ and $b$ are positive dimensional constants. The reference point for zero potential is chosen at the origin $(0,0,0)$. Which of the following statements is/are CORRECT?

Professor PC Thomas Classes · Answer

A is Correct: Using $E_x = -\frac{\partial \varphi}{\partial x}$ and $E_y = -\frac{\partial \varphi}{\partial y}$, integrating $\vec{E}_1$ yields $\varphi_1 = -ax^2y + \frac{1}{3}ay^3$. 
B is Correct: Similarly, integrating the components of $\vec{E}_2$ gives $\varphi_2 = -axy - byz$. 
C is Correct: Work done by the conservative field is $W = -\Delta U = q(\varphi_{initial} - \varphi_{final})$. Here, $\varphi(1,1) = -a + a/3 = -2a/3$ and $\varphi(2,0) = 0$. Thus, $W = q(-2a/3 - 0) = -2aq/3$. 
D is Correct: Using Gauss's Law in differential form, $ho = \epsilon_0(
abla \cdot \vec{E})$. For $\vec{E}_1$, the divergence is $\frac{\partial}{\partial x}(2axy) + \frac{\partial}{\partial y}(a(x^2-y^2)) = 2ay - 2ay = 0$.