## Electric Field And Electric Potential

**Electric Field And Electric Potential** Assignment Help | **Electric Field And Electric Potential** Homework Help

# 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. ThendV =

__∂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

For more help in

**Relation Between Electric Field and Electric Potential**please click the button below to submit your homework assignment.