Electric field Vector The Hub Of Electrostatics 1

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A user identified a pattern in electrostatics and shared their revision notes for feedback. They correctly stated that electric flux is defined by the equation φ = ∫E·dS. However, they mistakenly claimed that the electric field E can be expressed as the gradient of the electric flux, which is inaccurate. The correct relationship is that the electric field is the gradient of the electric potential, not the flux. Clarifications on these concepts are essential for accurate understanding in electrostatics.
i_am_ayush
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Hey I cought out a pattern in Electrostatics and i created it in my revision notes. I have uploaded it
please Suggest if any correction required
 

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You've written that the electric flux is given by,

\varphi = \int\vec{E}\cdot d\vec{S}

which is correct, but then you've written that

\vec{E} = -\vec{\nabla} \varphi which is wrong! The electric field can be written as the gradient of the electric potential, not the the flux.
 
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