Dipole (electrodynamics)

Earth's magnetic field approximately coincides with the dipole field. However, the “N” and “S” (north and south) poles are marked “geographically”, that is, the opposite of the accepted designation for the poles of a magnetic dipole.

A dipole is an idealized system that serves for an approximate description of the field created by more complex charge systems, as well as for an approximate description of the action of an external field on such systems. The dipole approximation , the fulfillment of which is usually implied when speaking of the dipole field , is based on expanding the field potentials in a series in powers of the radius vector characterizing the position of the source charges and discarding all terms above the first order ^[1] . The resulting functions will effectively describe the field if:

the dimensions of the field-emitting system are small compared to the distances considered, so that the ratio of the characteristic size of the system to the length of the radius vector is small and it makes sense to consider only the first terms of the expansion of the potentials in a series;
the first-order term in the expansion is not equal to 0, otherwise you need to use the higher multipolarity approximation;
the equations consider potential gradients not higher than first order.

A typical example of a dipole is two charges of equal magnitude and opposite in sign, located at a distance from each other that is very small compared to the distance to the observation point. The field of such a system is completely described by the dipole approximation.

Content

1 System dipole moment
- 1.1 Electric dipole
- 1.2 Magnetic dipole
2 Field of an oscillating dipole
- 2.1 Field at close range (near field)
- 2.2 Dipole radiation (radiation in the wave zone or far zone)
3 See also
4 notes
5 Literature

System dipole moment

Equipotential surfaces of an electric dipole

Electric Dipole

Power lines of an electric dipole

An electric dipole is an idealized electron-neutral system consisting of point and equal in absolute value positive and negative electric charges .

In other words, an electric dipole is a combination of two equal in magnitude unlike point charges at a certain distance from each other.

Vector product ${\vec {l}},$ ${\ displaystyle {\ vec {l}},}$ carried out from a negative charge to a positive, by the absolute value of the charges $q\,,$ ${\ displaystyle q \ ,,}$ called the dipole moment: ${\vec {d}}=q{\vec {l}}.$ ${\ displaystyle {\ vec {d}} = q {\ vec {l}}.}$

In an external electric field ${\vec {E}}$ ${\ displaystyle {\ vec {E}}}$ moment of forces acts on an electric dipole ${\vec {d}}\times {\vec {E}},$ ${\ displaystyle {\ vec {d}} \ times {\ vec {E}},}$ which seeks to rotate it so that the dipole moment unfolds along the direction of the field.

The potential energy of an electric dipole in a (constant) electric field is $-{\vec {E}}\cdot {\vec {d}}.$ ${\ displaystyle - {\ vec {E}} \ cdot {\ vec {d}}.}$ (In the case of an inhomogeneous field, this means a dependence not only on the moment of the dipole — its magnitude and direction, but also on the location and location of the dipole).

Far from the electric dipole, its electric field strength decreases with distance $R$ ${\ displaystyle R}$ as $R^{-3},$ ${\ displaystyle R ^ {- 3},}$ i.e. faster than a point charge ( $E\sim R^{-2}$ ${\ displaystyle E \ sim R ^ {- 2}}$ )

Any generally electrically neutral system containing electric charges, in some approximation (i.e., in the dipole approximation itself ) can be considered as an electric dipole with a moment ${\vec {d}}=\sum _{i}q_{i}{\vec {r}}_{i},$ ${\ displaystyle {\ vec {d}} = \ sum _ {i} q_ {i} {\ vec {r}} _ {i},}$ Where $q_{i}$ ${\ displaystyle q_ {i}}$ - charge $i$ ${\ displaystyle i}$ th element ${\vec {r}}_{i}$ ${\ displaystyle {\ vec {r}} _ {i}}$ Is its radius vector. In this case, the dipole approximation will be correct if the distance at which the electric field of the system is studied is large compared to its characteristic dimensions.

In the point approximation, the field created by a dipole at a point with a radius vector ${\vec {r}}$ ${\ displaystyle {\ vec {r}}}$ is given by the following relation:

{\vec {E}}={\frac {1}{4\pi \varepsilon _{0}}}{\frac {3{\vec {r}}({\vec {r}},{\vec {d}})-{r^{2}}{\vec {d}}}{r^{5}}}

{\ displaystyle {\ vec {E}} = {\ frac {1} {4 \ pi \ varepsilon _ {0}}} {\ frac {3 {\ vec {r}} ({\ vec {r}}, {\ vec {d}}) - {r ^ {2}} {\ vec {d}}} {r ^ {5}}}}

Magnetic Dipole

A magnetic dipole is an analogue of an electric one, which can be imagined as a system of two “magnetic charges” - magnetic monopoles . This analogy is conditional, since magnetic charges are not detected. As a model of a magnetic dipole, we can consider a small (in comparison with the distances at which the magnetic field generated by the dipole radiation) flat, closed conductive area frame $S\,,$ ${\ displaystyle S \ ,,}$ through which current flows $I\,.$ ${\ displaystyle I \ ,.}$ In this case, the magnetic moment of the dipole (in the GHS system) is the value ${\vec {\mu }}=IS{\vec {n}},$ ${\ displaystyle {\ vec {\ mu}} = IS {\ vec {n}},}$ Where ${\vec {n}}$ ${\ displaystyle {\ vec {n}}}$ - a unit vector directed perpendicular to the plane of the frame in the direction when observed, in which the current in the frame appears to be current clockwise.

Expressions for torque ${\vec {M}}$ ${\ displaystyle {\ vec {M}}}$ acting from the magnetic field to the magnetic dipole, and the potential energy of constant magnetic $U$ ${\ displaystyle U}$ dipoles in a magnetic field, similar to the corresponding formulas for the interaction of an electric dipole with an electric field, only the magnetic moment ${\vec {m}}$ ${\ displaystyle {\ vec {m}}}$ and magnetic induction vector ${\vec {B}}$ ${\ displaystyle {\ vec {B}}}$ :

{\vec {M}}={\vec {m}}\times {\vec {B}},

{\ displaystyle {\ vec {M}} = {\ vec {m}} \ times {\ vec {B}},}

U=-{\vec {m}}\cdot {\vec {B}}.

{\ displaystyle U = - {\ vec {m}} \ cdot {\ vec {B}}.}

Oscillating Dipole Field

This section discusses the field created by a point electric dipole. $\mathbf {d} (t),$ ${\ displaystyle \ mathbf {d} (t),}$ located at a given point in space.

Close Field ( Near Field)

The field of a point dipole oscillating in vacuum has the form

\mathbf {E} ={\frac {3\mathbf {n} (\mathbf {n} ,\mathbf {d} )-\mathbf {d} }{R^{3}}}+{\frac {3\mathbf {n} (\mathbf {n} ,{\dot {\mathbf {d} }})-{\dot {\mathbf {d} }}}{cR^{2}}}+{\frac {\mathbf {n} (\mathbf {n} ,{\ddot {\mathbf {d} }})-{\ddot {\mathbf {d} }}}{c^{2}R}}

{\ displaystyle \ mathbf {E} = {\ frac {3 \ mathbf {n} (\ mathbf {n}, \ mathbf {d}) - \ mathbf {d}} {R ^ {3}}} + {\ frac {3 \ mathbf {n} (\ mathbf {n}, {\ dot {\ mathbf {d}}}) - {\ dot {\ mathbf {d}}}} {cR ^ {2}}} + { \ frac {\ mathbf {n} (\ mathbf {n}, {\ ddot {\ mathbf {d}}} - {\ ddot {\ mathbf {d}}}} {c ^ {2} R}}}

\mathbf {B} =\left[{\frac {\dot {\mathbf {d} }}{cR^{2}}}+{\frac {\ddot {\mathbf {d} }}{Rc^{2}}},\mathbf {n} \right]=\left[\mathbf {n} ,\mathbf {E} +{\frac {\mathbf {d} }{R^{3}}}\right],

{\ displaystyle \ mathbf {B} = \ left [{\ frac {\ dot {\ mathbf {d}}} {cR ^ {2}}} + {\ frac {\ ddot {\ mathbf {d}}} { Rc ^ {2}}}, \ mathbf {n} \ right] = \ left [\ mathbf {n}, \ mathbf {E} + {\ frac {\ mathbf {d}} {R ^ {3}}} \ right],}

Where $\mathbf {n} ={\frac {\mathbf {R} }{R}}$ ${\ displaystyle \ mathbf {n} = {\ frac {\ mathbf {R}} {R}}}$ Is the unit vector in the considered direction, $c$ ${\ displaystyle c}$ Is the speed of light.

These expressions can be given a slightly different shape by introducing the Hertz vector

\mathbf {Z} =-{\frac {1}{R}}\cdot \mathbf {d} \left(t-{\frac {R}{c}}\right).

{\ displaystyle \ mathbf {Z} = - {\ frac {1} {R}} \ cdot \ mathbf {d} \ left (t - {\ frac {R} {c}} \ right).}

Recall that the dipole rests at the origin, so that $\mathbf {d}$ ${\ displaystyle \ mathbf {d}}$ is a function of one variable. Then

\mathbf {E} =-\operatorname {rot} \,\operatorname {rot} \,\mathbf {Z} ,

{\ displaystyle \ mathbf {E} = - \ operatorname {rot} \, \ operatorname {rot} \, \ mathbf {Z},}

\mathbf {B} =-{\frac {1}{c}}\operatorname {rot} \,{\dot {\mathbf {Z} }}.

{\ displaystyle \ mathbf {B} = - {\ frac {1} {c}} \ operatorname {rot} \, {\ dot {\ mathbf {Z}}}.}

In this case, the field potentials can be selected in the form

\mathbf {A} =-{\frac {\dot {\mathbf {Z} }}{c}},~~\phi =\operatorname {div} \,\mathbf {Z} .

{\ displaystyle \ mathbf {A} = - {\ frac {\ dot {\ mathbf {Z}}} {c}}, ~~ \ phi = \ operatorname {div} \, \ mathbf {Z}.}

These formulas can always be used when the dipole approximation is applicable.

Dipole radiation (radiation in the wave zone or far zone )

The above formulas are greatly simplified if the dimensions of the system are much less than the length of the emitted wave, that is, the speed of charges is much less than c , and the field is considered at distances much larger than the wavelength. This area of the field is called the wave zone . The propagating wave can be considered almost flat in this region. Of all the members in the expressions for $\mathbf {E}$ ${\ displaystyle \ mathbf {E}}$ and $\mathbf {B}$ ${\ displaystyle \ mathbf {B}}$ only members containing the second derivatives of $\mathbf {d} ,$ ${\ displaystyle \ mathbf {d},}$ as

{\frac {\dot {\mathbf {d} }}{c}}\approx {\frac {d}{\lambda }},

{\ displaystyle {\ frac {\ dot {\ mathbf {d}}} {c}} \ approx {\ frac {d} {\ lambda}},}

{\frac {\ddot {\mathbf {d} }}{c^{2}}}\approx {\frac {d}{\lambda ^{2}}}.

{\ displaystyle {\ frac {\ ddot {\ mathbf {d}}} {c ^ {2}}} \ approx {\ frac {d} {\ lambda ^ {2}}}.}

The expressions for the fields in the GHS system take the form

\mathbf {H} ={\frac {1}{c^{2}R}}[{\ddot {\mathbf {d} }},\mathbf {n} ],~~\mathbf {H} =[\mathbf {n} ,\mathbf {E} ],

{\ displaystyle \ mathbf {H} = {\ frac {1} {c ^ {2} R}} [{\ ddot {\ mathbf {d}}}, \ mathbf {n}], ~~ \ mathbf {H } = [\ mathbf {n}, \ mathbf {E}],}

\mathbf {E} ={\frac {1}{c^{2}R}}\left[[{\ddot {\mathbf {d} }},\mathbf {n} ],\mathbf {n} \right],~~\mathbf {E} =[\mathbf {B} ,\mathbf {n} ].

{\ displaystyle \ mathbf {E} = {\ frac {1} {c ^ {2} R}} \ left [[{\ ddot {\ mathbf {d}}}, \ mathbf {n}], \ mathbf { n} \ right], ~~ \ mathbf {E} = [\ mathbf {B}, \ mathbf {n}].}

In a plane wave , the radiation intensity into a solid angle $d\Omega$ ${\ displaystyle d \ Omega}$ is equal to

dI=c{\frac {H^{2}}{4\pi }}R^{2}d\Omega ,

{\ displaystyle dI = c {\ frac {H ^ {2}} {4 \ pi}} R ^ {2} d \ Omega,}

therefore for dipole radiation

dI={\frac {1}{4\pi c^{3}}}[{\ddot {\mathbf {d} }},\mathbf {n} ]^{2}d\Omega ={\frac {{\ddot {\mathbf {d} }}^{2}}{4\pi c^{3}}}\sin ^{2}{\theta }d\Omega .

{\ displaystyle dI = {\ frac {1} {4 \ pi c ^ {3}}} [{\ ddot {\ mathbf {d}}}, \ mathbf {n}] ^ {2} d \ Omega = { \ frac {{\ ddot {\ mathbf {d}}} ^ {2}} {4 \ pi c ^ {3}}} \ sin ^ {2} {\ theta} d \ Omega.}

Where $\theta$ ${\ displaystyle \ theta}$ - angle between vectors ${\ddot {\mathbf {d} }}$ ${\ displaystyle {\ ddot {\ mathbf {d}}}}$ and $\mathbf {n} .$ ${\ displaystyle \ mathbf {n}.}$ Find the total radiated energy. Given that $d\Omega =2\pi \,\sin {\theta }\,d\theta ,$ ${\ displaystyle d \ Omega = 2 \ pi \, \ sin {\ theta} \, d \ theta,}$ integrate the expression over $d\theta$ ${\ displaystyle d \ theta}$ from $0$ ${\ displaystyle 0}$ before $\pi .$ ${\ displaystyle \ pi.}$ Total emission is equal to

I={\frac {2}{3c^{3}}}{\ddot {\mathbf {d} }}^{2}.

{\ displaystyle I = {\ frac {2} {3c ^ {3}}} {\ ddot {\ mathbf {d}}} ^ {2}.}

We indicate the spectral composition of the radiation. It is obtained by replacing the vector ${\ddot {\mathbf {d} }}$ ${\ displaystyle {\ ddot {\ mathbf {d}}}}$ by its Fourier component and at the same time multiplying the expression by 2. Thus,

d{\mathcal {E}}_{\omega }={\frac {4\omega ^{4}}{3c^{3}}}\left|\mathbf {d} _{\omega }\right|^{2}{\frac {d\omega }{2\pi }}.

{\ displaystyle d {\ mathcal {E}} _ {\ omega} = {\ frac {4 \ omega ^ {4}} {3c ^ {3}}} \ left | \ mathbf {d} _ {\ omega} \ right | ^ {2} {\ frac {d \ omega} {2 \ pi}}.}

Notes

↑ For the case of electrostatics, magnetostatics, etc., this means that the potential with the powers of the radius vector from the dipole to the observation point is -1 and -2; in the case of a purely dipole field (when the source system has a zero total charge) only of degree -2.

Literature

Landau L.D. , Lifshits E.M. Field Theory. - 7th edition, revised. - M .: Nauka , 1988 .-- 512 p. - (“ Theoretical Physics ”, Volume II). - ISBN 5-02-014420-7 .
Akhmanov S. A., Nikitin S. Yu. , “Physical Optics”, 2004.

[1] For the case of electrostatics, magnetostatics, etc., this means that the potential with the powers of the radius vector from the dipole to the observation point is -1 and -2; in the case of a purely dipole field (when the source system has a zero total charge) only of degree -2.