1.5. Connection between intensity and potential.

Intensity and potential are various characteristics of one point of a field.

We will consider work of electric forces in electric field when electric is moving from the point 1 to the point 2.

A = QExΔx;

A = Q (φ1-φ2) = - QΔφ

Having equated both expressions for work, we will receive: QEx∆x=- Q∆φ,

E

α 2 E

1

x

fig.1.5.

.

Similarly, ,

.

Hence, Е= - gradφ.

The intensity at any point of the field is equal to the rate of potential gradient in this point of the field, taken with the opposite sign. The minus sign indicates that the intensity vector is directed toward decreasing of the potential, i.e. intensity and potential vectors are oppositely directed.

1.6. Equipotential surfaces

Imaginary surfaces, all points of which have the same potential are called equipotential surfaces. Their equations are:

φ(x, y ,z)=const.

When moving through the equipotential surface by dl, the potential φ is not changed (dφ=0). Then the component of the intensity vector, which is tangent to the surface, equals zero. Thus, the intensity vector at each point is directed along the normal line to the equipotential surface which passes through a given point. Hence, electric field lines at each point are orthogonal to an equipotential surface.

Equipotential surface can be made through any point of the field; there can be a set of such surfaces. These surfaces are made in a way that the potential difference for the two adjacent surfaces should be everywhere one and the same. In this case, the density of the equipotential surfaces can indicate the magnitude of the field intensity. The denser equipotential surfaces are, the faster the potential gradient changes when moving along a normal line to the surface, respectively, the more andЕ are in this location.

For a homogeneous field, equipotential surfaces are a system of equidistant planes perpendicular to the direction of the field.

For a point charge, equipotential surfaces can be represented as shown in figure 1.6.

fig.1.6.

1.7. Electric dipole

Electric dipole is a system of two, equal in magnitude, dissimilar point charges +q and -q, the distance l between which is much less than the distance to those points, in which the field of the system is defined. The line passing through both charges is called the dipole axis; l – dipole arm.

Dipole field has axial symmetry. If the distance between the charges does not change, then this dipole is called rigid. If the length of the dipole arm l is small compared with the distance r to the observation point, then this dipole is called point.

The main characteristic of electric dipole is its electric dipolar moment p – a vector which is numerically equal to the product of a charge and an arm, and is directed from the negative charge to the positive one.

+q r

l -q

fig.1.7

1.8. Electric dipole in a homogeneous external electric field

Let us consider the action of an external electric field on a dipole.

If the field is homogeneous, the forces acting on the positive and negative dipole charges are equal in magnitude and opposite in direction, that is, form a pair of forces (Figure 1.8). Accordingly, their total force equals zero.

The action of force pair is characterized by a moment of couple:

M= qElsinα,

α – an angle between the vector l and the field intensity Е.

, then

М=plsinα.

Or, in vector form

.

+q F

l E

F α

-q

fig.1.8

Thus, in a homogeneous electric field, a dipole is affected by a force pair, which is trying to regain the dipole so that the angle between the vectors p and E decreases and the dipole takes the field direction.

There are two equilibrium points of a dipole:

• a dipole is parallel to the electric field (stable equilibrium);

• a dipole is antiparallel to it (unstable equilibrium).

The energy of a dipole in a homogeneous electric field with intensity Е:

.

10. Dipole in non-homogeneous external electrical field

If the field is inhomogeneous, the forces F₁ and F₂ are different in magnitude and their total force does not equal zero.

Let’s find the total force. Consider as the dipole is located along one of the lines of force (Fig. 1.9). Then .

Thus, in an inhomogeneous electric field, a dipole is affected except the moment of a force pair, by a force in the direction of increasing field strength, tending to draw the dipole into the strong field region.

fig. 1.9

Lecture 2