How is electric field at a point related to potential gradient?

Explanation

Let a charge + q is placed at O. Consider two points A and B in its electric field. The points A and B are so close that electric field intensity at them remains same.

Let a test charge `q_(0)` is placed at point A.
`F=q_(0)E=` Force acting on go along AX.
Small amount of work done in moving `q_(0)` from A to B (AB = dr) is given as
`dW=vecF.vec(dr)" "[:.vecFand vec(dr) "are in opposite directions"]`
`dW=-Fdrcos180^(@)`
`dW=-Fdr`
`dW=-q_(0)Edr" "[:.F=q_(0)E]`
`(dW)/(q_(0))=-Edr`
We know `(dW)/(q_(0))=dV`= potential difference
`impliesdV=-Edr`
or `E=-(dV)/(dr)`
It means electric field intensity at a point is equal to the negative of potential gradient at that point.