A

To perform the calculations, we first need to identify the terms in the equation
1. The equation describes a physical relationship involving electric fields, where we have \( E = 2y \hat{i} + 2x \hat{j} \).
2. The components of the electric field are: \( E_x = 2y \) and \( E_y = 2x \).
3. To find the scalar potential function \( V(x, y, z) \) corresponding to this electric field, we recognize that \( E = -\nabla V \).
4. Integrate the components:

• From \( E_x = -\frac{\partial V}{\partial x} \Rightarrow V = -\int E_x \, dx + f(y,z) = -\int 2y \, dx + f(y,z) = -2xy + f(y,z)
• From \( E_y = -\frac{\partial V}{\partial y} \Rightarrow V = -\int E_y \, dy + g(x,z) = -\int 2x \, dy + g(x,z) = -2xy + g(x,z)

Comparing both results leads to: \( f(y,z) = g(x,z) = h(z) \). Thus, \( V(x, y, z) = -2xy + h(z) \).
5. Therefore, we find that \( V(x, y, z) \) depends on \( z \) through an arbitrary function that is constant with respect to \( x \) and \( y \). Hence, the result depends heavily on interpretation of the relation given in the problem. Thus, we conclude with \( V(x, y, z) = -2xy + C \), where \( C \) is a constant.