Question

Establish a relation between electric field (E) and potential difference (V) between two points in the field.

Answer 1

Consider two points A and B with a small distance dr between them. If q is the charge placed at point B and moves towards point A then some work must be done and displacement will be in opposite direction as compared to the direction of electric field. Here the angle between force and displacement will be 180°.
Since, the distance between points tends to zero, so force will be same at both points.
At point B, force on charge q = qE
At point A, force on charge q = qE
The potential difference between two points A and B will be
V₍B₎ - V₍A₎ - dV = V = V₍AB₎ = W₍AB₎ / q
V₍AB₎ = Force.Displacement.cosΘ/q
V₍AB₎ = F.dr.cos180°/q
V₍AB₎ = -Fdr/q
V₍AB₎ = - qEdr/q
V₍AB₎ = -Edr
So, dV = -Edr
E = - dV/dr
This equation gives the relation between E and dV. Negative sign shows that potential difference decreases in the direction of electric field.