If the electric potential at a point P(x, y) | vedclass

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(A) The electric field $\vec{E}$ is related to the potential $V$ by the relation $\vec{E} = -
abla V$.Given $V = axy$,the components of the electric

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$r$

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(A) The electric field $\vec{E}$ is related to the potential $V$ by the relation $\vec{E} = -
abla V$.Given $V = axy$,the components of the electric field are:$E_x = -\frac{\partial V}{\partial x} = -ay$$E_y = -\frac{\partial V}{\partial y} = -ax$The magnitude of the electric field is $E = \sqrt{E_x^2 + E_y^2}$.Substituting the components,we get $E = \sqrt{(-ay)^2 + (-ax)^2} = \sqrt{a^2y^2 + a^2x^2} = a\sqrt{x^2 + y^2}$.Since the distance $r$ from the origin is defined as $r = \sqrt{x^2 + y^2}$,we have $E = ar$.Therefore,the electric field $E$ is proportional to $r$ $(E \propto r)$.

If the electric potential at a point $P(x, y)$ is given by $V = axy$,then the electric field at a distance $r$ from the origin is proportional to:

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