Chapter 1

Electrostatics and Magnetostatics

Electrostatic dipole

DéfinitionElectrostatic dipole

A rigid set of two punctual charges and (globally neutral) at the distance from each other is called an “electrostatic diplole”.

Such a model can explain :

  • Polar molecules (for instance : HCl, H2O)

  • Atom polarization in an exterior electric field (solvation of ions phenomenon)

FondamentalDetermination of the potential created by the dipole

We calculate the potential created by the dipole located in a point M in space.

Revolution symmetry around (Oz) axis : we choose in the plane of the two charges.

Under the dipole approximation (we study the charges from « far » away, i.e. at a distance greatly superior to a few ).

Calculation of the electrostatic potential

The potential in the point M is (superposition principle) :

(With : )

According to the Chasles relation :

The squared value :

Then we compute  :

The Taylor expansion until the order is :

Let be such as ( ), then :

In the same manner (if is substituted by and by ) :

Then :

Thus the potential :

The dipole moment vector of the electrostatic dipole can be defined by (the vector is oriented from the negative charge to the positive one) :

Then (the potential decreases like ) :

By posing :

DéfinitionDipole moment vector

The dipole moment vector is a characteristic of the electrostatic dipole.

In the international system of units is expressed in .

Yet the Debye unit is more adapted :

Example :

A molecule of water has a dipole moment.

Deduce the charges and carried respectively by hydrogen and oxygen atoms.

We know (see figure) :

The total dipole moment is given by :

With :

Consequently :

Thus can be deduced :

Water molecule

MéthodeCalculation of the electrostatic field under dipole approximation

The intrinsic relation lets us calculate the field (in polar coordinates) :

Consequently :

Or even :

ComplémentDipole field topography

  • Equipotential surface :

    The equipotential surface ( ) equation at the potential is :

    Thus the equation in polar coordinates :

    The aspect of equipotential lines is shown on the following picture.

    The axis perpendicular to (Oz) and passing through point O is the zero equipotential.

    By rotating around (Oz), these lines create equipotential surfaces.

Dipole field topography
  • Field lines :

    On the field line passing through point M :

    Or :

    Thus :

    And :

Field lines

By replacing the field coordinates by their expressions :

Let's separate variables :

Then :

Let be for . Then, by integration, we obtain the equation of the field lines in polar coordinates ( is a parameter) :

The aspect of field lines is given on the previous picture.

FondamentalAction of an exterior electric field on a dipole

Let an electrostatic dipole be immersed in an electric field which can be supposed uniform at the dipole scale.

What effect does the field have on the dipole ?

  • System studied : the (rigid) dipole

  • Referential of study : laboratory referential (supposed Galilean)

Torque on dipole

The dipole undergoes two forces and which addition is equal to zero.

The dipole undergoes a “force torque”, which moment in relation to O is :

Thus :

Finally :

Under the effect of an electric field, the dipole starts spinning in order to align with the direction of the field ( and in the same orientation, stable equilibrium position).

Energetic study :

Let be the potential from which the field ; the potential energy of the (rigid) dipole in this field is :

We remind that is a gradient field, so :

Thus, with :

Then :

So :

is minimum (stable equilibrium position) when and are oriented in the same way.

The following picture illustrates the solvation phenomenon : water molecules align with the field lines created by the charge .

Ion solvation
Vector calculus