Relation between electric field and potential difference

Relation between Electric Field and Potential Difference:

Consider an arbitrary curve AB in a non-uniform electric field [in Fig]. Let P be any point on this curve. We know that this potential difference between A and B is equal to the negative line integral of field strength from A to B. 

Motion of a point charge in a non-uniform electric field. 

∴     V(rB) - V(rA) = - BA∫ E(r) . dl

If the point A is removed to infinity, potential at any point is given by

        V(r) = - r∞∫ E(r). dl

Let Q be another point situated very close to P at a very small distance 'dl', so that the field between P and Q is practically the same. In that case the potential difference 'dV' between these points can be written as

      dV = - E. dl

            = - E dl cos θ

Where  'θ' is the angle between E and dl

∴      dV = - E cos θ dl      or     dV = - ET dl

Where  represents the tangential component of E along dl. 

∴     ET = - dV/dl

Here, -dV/dl denotes the rate of change of potential along a line and is known as the potential gradient. 

Therefore, the component of electric intensity, along any direction is equal to the negative potential gradient in that particular direction. 

The negative sign in equation ET = - dV/dl indicates that the direction of E coincides with that of decreasing potential.