The electric potential at a point $(x, y)$ in an electric field is given by $V = 6 x y + y^{2} - x^{2}$ Calculate the electric field at that point.

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Solution

$\begin{matrix} \to E = - \to \nabla V, \to E = - \frac{\partial V}{\partial x}^i - \frac{\partial V}{\partial y}^j - \frac{\partial V}{\partial z}^k \to V = [6 x y + y^{2} - x^{2}] \to E = - \frac{\partial}{\partial x} (6 x y + y^{2} - x^{2})^i - \frac{\partial}{\partial y} [6 x y + y^{2} - x^{2}] ˙ j - \frac{\partial}{\partial z} [6 x y + y^{2} - x^{2}]^k \to E = - [6 y - 2 x]^i - [6 x + 2 y]^j - 0 \to E = (2 x - 6 y)^i - (6 x + 2 y)^j v o l t / m \end{matrix}$

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