Electrostatics and Magnetostatics
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 ).
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 ) :
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 :
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 :
Thus can be deduced :
Méthode : Calculation of the electrostatic field under dipole approximation
The intrinsic relation lets us calculate the field (in polar coordinates) :
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 :
By replacing the field coordinates by their expressions :
Let's separate variables :
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)
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 :
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 :
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 .