Electrostatic dipole

A electrostatic dipole is defined by a couple of 2 points P and NR of respective load +q and - Q distant of has .

This concept is mainly used in Chimie where certain connections between molecules can be explained by modelling these molecules by a dipole (Hydrogen bond for example).

In Physical, one is interested in the Electric field $\backslash \; vec\; \{E\}\; (M)$ creates in a point M far away from the dipole (one then speaks about active dipole ). But one can also study the behavior of the dipole when it is placed in an external field (one then speaks about passive dipole ).

Dipole moment

We will define a size which characterizes an electrostatic dipole: its dipole moment $\backslash \; vec\; \{p\}$ given by:

$\backslash \; vec\; \{p\}\; =q\; \backslash \; cdot\; \backslash \; overrightarrow\; \{NP\}$

(It is pointed out that the point P with the load +q , the point NR the load - Q .)

The dipole moment is expressed in Coulomb - Mètre (Cm); by measurement of convenience, one will express it in Debye (D):

1 D = 3,33564 × 10^-30 Cm

Active dipole

Generated potential

The potential V (M) creates by the dipole in M is given according to the law of Coulomb by:

$V\; (M)\; =\; \backslash \; frac\; \{Q\}\; \{4\; \backslash \; pi\; \backslash \; epsilon\_0\}\; \backslash \; left\; (\backslash \; frac\; \{1\}\; \{PM\}\; -\; \backslash \; frac\; \{1\}\; \{Nm\}\; \backslash \; right)$

It is supposed moreover that the point M is very far away from the dipole. If one places in polar Coordonnées with for origin O medium of and for polar axis line (PM) (cf diagram on the right), this assumption results in $r\; \backslash \; gg\; a$, and one has then by a Développement limited:

$\backslash \; frac\; \{1\}\; \{PM\}\; -\; \backslash \; frac\; \{1\}\; \{Nm\}\; =\; \backslash \; frac\; \{has\; \backslash \; cdot\; \backslash \; cos\; \backslash \; theta\}\; \{r^2\}$

The following formula then is obtained:

$V\; (M)\; =\; \backslash \; frac\; \{Q\; \backslash \; cdot\; has\; \backslash \; cdot\; \backslash \; cos\; \backslash \; theta\}\; \{4\; \backslash \; pi\; \backslash \; epsilon\_0\; r^2\}\; =\; \backslash \; frac\; \{\backslash \; vec\; \{p\}\; \backslash \; cdot\; \backslash \; vec\; \{R\}\}\; \{4\; \backslash \; pi\; \backslash \; epsilon\_0\; r^3\}$

The equipotential S far from the dipole are then given by the equation $r=k\; \backslash \; sqrt\; \{\backslash \; cos\; \backslash \; theta\}$ (cf graph on the right).

Generated electric field

The knowledge of V (M) enables us to deduce directly $\backslash \; vec\; \{E\}\; (M)$ by the formula:

$\backslash \; vec\; \{E\}\; (M)\; =\; -\; \backslash \; overrightarrow\; \{\backslash \; rm\; grad\}\; \backslash \; bigl\; (V\; (M)\; \backslash \; bigr)\; =\; \backslash \; frac\; \{p\}\; \{4\; \backslash \; pi\; \backslash \; epsilon\_0r^3\}\; \backslash \; cdot\; \backslash \; bigl\; (2\; \backslash \; cos\; \backslash \; theta\; \backslash \; vec\; \{u\_r\}\; +\; \backslash \; sin\; \backslash \; theta\; \backslash \; vec\; \{u\_\; \backslash \; theta\}\; \backslash \; bigr)$

From where the formula:

$\backslash \; vec\; \{E\}\; (M)\; =\; \backslash \; frac\; \{1\}\; \{4\; \backslash \; pi\; \backslash \; epsilon\_0\; \backslash \; cdot\; r^3\}\; \backslash \; cdot\; \backslash \; bigl\; (3\; \backslash \; vec\; \{u\_r\}\; (\backslash \; vec\; \{p\}\; \backslash \; cdot\; \backslash \; vec\; \{u\_r\})\; -\; \backslash \; vec\; \{p\}\; \backslash \; bigr)$

One from of deduced the lines from field (orthogonal with equipotential) far from the dipole : $r\; =\; K\; \backslash \; sin^2\; \backslash \; theta$ (cf graph on the right).

Passive dipole

That is to say a dipole of dipole moment $\backslash \; vec\; \{p\}$ placed in M where reign an external field $\backslash \; vec\; \{E\}\; (M)$.

It is shown whereas the dipole undergoes a electrostatic Force $\backslash \; vec\; \{F\_e\}$ whose expression is given by: $\backslash \; vec\; \{F\_e\}\; =\; \backslash \; left\; (\backslash \; vec\; \{p\}\; \backslash \; cdot\; \backslash \; overrightarrow\; \{\backslash \; rm\; grad\}\; \backslash \; right)\; \backslash \; vec\; \{E\}\; (M)$

It undergoes also a moment $\backslash \; vec\; \{\backslash \; Gamma\_e\}\; (M)$ given by: $\backslash \; vec\; \{\backslash \; Gamma\_e\}\; (M)\; =\; \backslash \; vec\; \{p\}\; \backslash \; wedge\; \backslash \; vec\; \{E\}\; (M)$

Lastly, if the axis of the dipole cannot turn, one can associate an electrical potential energy to him $E\_p=-\; \backslash \; vec\; \{p\}\; \backslash \; cdot\; \backslash \; vec\; \{E\}\; (M)$

See too

• Electrostatic
• electric Dipole of a ball to compare with magnetic Dipole of a sphere
• Quadrupole
• Polarization
• Polarity
• magnetic Moment

Random links:

He was born the divine infant | Sporting union montalbanaise | Aurélie Felix | Rachel York | Route main road 8bis | Brigade_droite_de_citoyens

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org