Chapter 1

# Electrostatic dipole

### Définition : Electrostatic 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)

### Fondamental : Determination 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 : ### Définition : Dipole 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

### Méthode : Calculation of the electrostatic field under dipole approximation

The intrinsic relation lets us calculate the field (in polar coordinates) : Consequently : Or even : ### Complément : Dipole 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.

• 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.

### Fondamental : Action 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 .

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