Chapter 2.3, Problem 2.20P

One of these is an impossible electrostatic field. Which one?
(a) $E=k\left[xy\text{<mover accent="true"> <mo> x </mo> <mo> ^ </mo> </mover> +2}yz\text{<mover accent="true"> <mo> y </mo> <mo> ^ </mo> </mover>}+3xz\text{<mover accent="true"> <mo> z </mo> <mo> ^ </mo> </mover>}\right];$
(b) $E=k\left[y^2\text{<mover accent="true"> <mo> x </mo> <mo> ^ </mo> </mover> +}\left(2xy+z^2\right)\text{<mover accent="true"> <mo> y </mo> <mo> ^ </mo> </mover>}+2yz\text{<mover accent="true"> <mo> z </mo> <mo> ^ </mo> </mover>}\right].$
Here k is a constant with the appropriate units. For the possible one, find the potential, using the origin as your reference point. Check your answer by computing $\nabla V$ . [Hint: You must select a specific path to integrate along. It doesn’t matter what pathyou choose, since the answer is path-independent, but you simply cannot integrateunless you have a definite path in mind.]