Relation Between Electric Field and Electric Potential

We can calculate the electric field E if potential function V is known throughout a certain region of space. Consider two neighboring points A (x, y, z) and B (x + dx, y + dy, z + dz) distance dl apart in the region. Let the value of potential at A and B be V and V + dV respectively. Let dV be the change in potential in going from A to B. Then

dV = ∂V dx + ∂V dy + ∂V dz
         ∂x           ∂y         ∂z

       = ( i ∂V + j ∂V + k ∂V )
               ∂x       ∂y       ∂z

     (i dx + j dy + k dz)
Here, i dx + j dy + k dz is the displacement vector dl between A and B.
Thus
    dV = (grade V). dl

The work done by the external agent in moving a test charge q from A to B along dl is
    dW = F . dl = - q E . dl.
or    dW/q = - E . dl

But, by definition, dW/q is the potential difference dv between the points A and B. Thus
    dV = - E . dl.

Comparing Eqs. (1) and (2),
    E = - grade V = - Δ V.

Thus the electric field at any point is the negative of the gradient of potential at that point. The minus sign indicates that E point in the direction of decreasing V.

Let Ex, Ey and Ez be the components of E along x, y and z axes. Then

    Ex = – ∂V, Ey  = – ∂V, Ez = – ∂V .
                ∂x    ____  ∂y __        ∂z