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Electric Potential at a Point in Space Due to a Point Charge Q

In a certain ...

Question

In a certain region the electric potential is given by the formula V(x, y, z) $= 2 x 2 y + 2 y 3 z - 4 z 4 x$ . Find the components of electric field and the vector electric field at point $(1, 1, 1)$ in this field.

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Solution

If expression of eleectric potential $(V)$ is given then electric field $(\to E)$ can be calculated by $\frac{\partial V_{x}}{\partial x}^i$ + $\frac{\partial V_{y}}{\partial y}^j$ + $\frac{\partial V_{z}}{\partial z}^k$
So, $\to E =$ $\frac{\partial (2 x 2 y)}{\partial x}^i$ + $\frac{\partial (2 y 3 z)}{\partial y}^j$ $- \frac{\partial (4 z 4 x)}{\partial z}^k$ $= 4 y^i + 6 z^j - 16 z^k$
At point $(1, 1, 1)$ , $\to E = 4^i + 6^j - 16^k$

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