The electric potential at a point (x, y, z) is given by V = −(x^2)y − x(z^3) + 4. The electric field E at that point is ?

The electric potential at a point \( (x, y, z) \) is given by \( V = -x^2 y - xz^3 + 4 \). The electric field \( \vec{E} \) at that point is ?

(a) \( \vec{E} = \hat{i}(2xy) + \hat{j}(x^2 + y^2) + \hat{k}(3xz - y^2) \)

(b) \( \vec{E} = \hat{i}(z^3) + \hat{j}(xyz) + \hat{k}(z^2) \)

(c) \( \vec{E} = \hat{i}(2xy - z^3) + \hat{j}(xy^2) + \hat{k}(3z^2x) \)

(d) \( \vec{E} = \hat{i}(2xy + z^3) + \hat{j}(x^2) + \hat{k}(3xz^2) \)

Answer :  (d) \( \vec{E} = \hat{i}(2xy + z^3) + \hat{j}(x^2) + \hat{k}(3xz^2) \)

\( \vec{E} = - \left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right) \)

\( = - \left[ (-2xy - z^3)\hat{i} + (-x^2)\hat{j} + (-3xz^2)\hat{k} \right] \)