Electric potential at a point (x, y, z) is given by V=-x^2y-xz^3 +4 . electric field at that po

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2. Electric Potential and Capacitance

easy

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

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

$\;\vec E = {z^3}\hat i + xyz\hat j + {z^2}\hat k$

$\;\vec E = \left( {2xy - {z^3}} \right)\hat i + x{y^2}\hat j + 3{z^2}x\hat k$

$\;\vec E = \left( {2xy + {z^3}} \right)\hat i + {x^2}\hat j + 3x{z^2}\hat k$

(AIPMT-2009)

Solution

The electric potential at a point,

$V=-x^{2} y-x z^{3}+4.$

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

$\therefore \vec{E}=\hat{i}\left(2 x y+z^{3}\right)+\hat{j} x^{2}+\hat{k}\left(3 x z^{2}\right)$

Standard 12

Physics

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

Solution

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