# dipole

Vector of a dipole consisting of two opposite charges of any kind.

A dipole (Greek: prefix di- : two-, πόλος (pólos) = "axis") is the physical arrangement of two opposing general charges, for example electrical charges or, in the case of a magnetic dipole, the exit surfaces of the magnetic field from a body. Since the opposite charges compensate each other, the dipole has no charge overall. The dipole is characterized by the distance and the amount of opposite charges . The product of these two quantities is the dipole moment in the multipole expansion of its far field. In this view, the actual dipole can be replaced by an expansionless dipole with the same dipole moment, which is located in its center (so-called “dipole limes”). ${\ displaystyle \ textstyle {\ vec {d}}}$${\ displaystyle \ textstyle q}$${\ displaystyle \ textstyle {\ vec {p}}: = {\ vec {d}} \ cdot q}$

For example, a dipole can be generated from electrical charges, but it can also exist without spatially separable charges, as in the case of a magnetic dipole (there are only fictitious , no real magnetic charges!).

In addition to electromagnetism , dipoles appear in various other areas such as acoustics or fluid dynamics . The directional dependence and the decrease in the generated field with distances are always characteristic . ${\ displaystyle \ textstyle {\ frac {1} {r ^ {3}}}}$${\ displaystyle \ textstyle r \ gg \ left \ vert {\ vec {p}} \ right \ vert}$

The concept of the dipole is not identical in its meaning to that of the two-pole , which describes a certain group of electrical circuits .

## Occurrence

### Electric dipoles

Electric dipoles require the separation of charges and therefore rarely occur on the macroscopic scale . On the microscopic scale , however, electric dipoles are very common. For example, they are made of asymmetrical molecules such as. B. the water molecule .

Electrical dipole moments also arise in biological muscle and nerve fibers as a result of built-up voltages, which can be measured, for example, in the electrocardiogram .

### Magnetic dipoles

Magnetic dipole field of the earth

Due to the lack of real magnetic monopoles , magnetic fields always originate from magnetic dipoles and their superimpositions. Therefore, in magnetism, macroscopically obvious dipole fields are very common. A long bar magnet can be described as a magnetic dipole to a good approximation. The earth's magnetic field also resembles a dipole field with a dipole axis from north to south.

A magnetic dipole generally arises from an area around which current flows or is connected to the spin of particles .

Larger configurations are also referred to as dipole magnets whose field is not a pure dipole field, but is similar to it, in contrast to quadrupole magnets and even higher orders of multipole expansion .

### Temporally variable dipoles

A static dipole field decreases ( : distance). For large distances, the enclosed surface increases with it, but the product also approaches zero. The electrical flow of the dipole field therefore disappears at a great distance . This also follows directly from the perception: from a great distance the poles can no longer be spatially distinguished, their field contributions cancel each other out. ${\ displaystyle \ sim 1 / r ^ {3}}$${\ displaystyle r}$${\ displaystyle \ sim r ^ {2}}$${\ displaystyle \ sim 1 / r}$

Time-variable dipoles behave fundamentally differently. Only they enable distant stars to be seen in the sky and the sun to supply the earth with radiant energy. A mathematical model of a simple variable dipole is the Hertzian dipole . Systems with dimensions in the order of magnitude of the wavelength are called dipole antennas .

## Physical description

Each dipole is characterized by its dipole moment, a vector quantity which has direction and magnitude . Here stands for an electrical dipole moment and in the following for any dipole moment, whereas a magnetic dipole moment is usually referred to as. ${\ displaystyle {\ vec {p}}}$${\ displaystyle {\ vec {m}}}$

### Physical dipole

A physical dipole consists of two opposite charges at a sufficiently short distance  d . The dipole moment is defined as ${\ displaystyle \ pm q}$

${\ displaystyle {\ vec {p}} = q \ cdot {\ vec {d}} \ ,.}$

${\ displaystyle {\ vec {d}}}$ shows from the negative to the positive charge.

The field at a great distance, i.e. H. for , then only depends on and no longer on q and d individually. The greater the distance, the closer the field approaches that of a point dipole. At small distances, the field deviates from this, which is also shown by non-vanishing higher multipole moments . ${\ displaystyle | r | \ gg d}$${\ displaystyle {\ vec {p}}}$

### Point dipole

Field lines of a point dipole

The point dipole arises when an extended dipole without a monopole moment is reduced to a point without changing the dipole moment. This corresponds to the borderline case with large distances and leads to charge distribution

${\ displaystyle \ rho ({\ vec {r}}) = - ({\ vec {p}} \ cdot {\ vec {\ nabla}}) \, \ delta ({\ vec {r}})}$

under use

• of the Nabla operator ${\ displaystyle {\ vec {\ nabla}}}$
• the delta function ${\ displaystyle \ delta ({\ vec {r}}).}$

A point dipole at the origin of the coordinate system generates the field potential

${\ displaystyle \ phi ({\ vec {r}}) = {\ frac {1} {4 \ pi \ varepsilon _ {0}}} {\ frac {{\ vec {p}} \ cdot {\ vec { r}}} {r ^ {3}}} = {\ frac {1} {4 \ pi \ varepsilon _ {0}}} {\ frac {p \ cdot \ cos (\ theta)} {r ^ {2 }}}}$

under use

• of the polar angle (measured from the dipole axis)${\ displaystyle \ theta}$
• the electric field constant ${\ displaystyle \ varepsilon _ {0}}$

and the vector field

${\ displaystyle {\ vec {E}} ({\ vec {r}}) = {\ frac {1} {4 \ pi \ varepsilon _ {0}}} \ left (3 \, {\ frac {{\ vec {p}} \ cdot {\ vec {r}}} {r ^ {5}}} \, {\ vec {r}} - {\ frac {1} {r ^ {3}}} \, { \ vec {p}} \ right) = {\ frac {1} {4 \ pi \ varepsilon _ {0}}} {\ frac {p} {r ^ {3}}} \ left (2 \ cos (\ theta) \ cdot {\ hat {r}} + \ sin (\ theta) \ cdot {\ hat {\ theta}} \ right)}$

under use

• of the unit vectors ${\ displaystyle {\ hat {v}}.}$

### Dipole in the multipole expansion

Fields that arise from a spatially limited charge distribution can be split up by the multipole expansion according to different proportions, which decrease at different speeds at large distances. In the case of large distances, the first non-vanishing term always dominates. The dipole term as the second term in the development is therefore particularly important when the monopole term (total charge) disappears. Any charge distribution then has the dipole moment

${\ displaystyle {\ vec {p}} = \ sum _ {i} q_ {i} \ cdot {\ vec {r}} _ {i}.}$

However, if the monopole term does not disappear, the value of the dipole moment can be changed by shifting the coordinate origin and is therefore not clearly defined.

The next higher term is the quadrupole moment , the field of which also decreases. ${\ displaystyle 1 / {r ^ {4}}}$

### Dipole in the external field

A dipole in an external field that is not generated by itself - ( electric field or magnetic field ) - has the potential energy : ${\ displaystyle {\ vec {E}}}$ ${\ displaystyle {\ vec {B}}}$

${\ displaystyle V = - {\ vec {p}} \ cdot {\ vec {E}}}$ or.
${\ displaystyle V = - {\ vec {m}} \ cdot {\ vec {B}}.}$

In an inhomogeneous external field, the force acts on a dipole :

${\ displaystyle {\ vec {F}} = {\ vec {\ nabla}} ({\ vec {p}} \ cdot {\ vec {E}})}$ or.
${\ displaystyle {\ vec {F}} = ({\ vec {\ nabla}} \ otimes {\ vec {B}}) {\ vec {m}}.}$

These two expressions are mathematically identical via the Graßmann identity when the magnetic field is free of rotation .

For this reason, one sometimes uses a slightly different, equivalent convention for the definition of the magnetic moment, namely

${\ displaystyle {\ vec {m}} _ {H}: = \ mu _ {0} \, {\ vec {m}}}$

with the magnetic field constant ${\ displaystyle \ mu _ {0}.}$

This is an alternative in the magnetic case

${\ displaystyle {\ vec {F}} = ({\ vec {\ nabla}} \ otimes {\ vec {H}}) {\ vec {m}} _ {H}}$

with the magnetic field strength . a. This is advantageous because the size of the magnetization of a permanent magnet , which is important in solid-state magnetism, has the same physical dimension as (and not as ). ${\ displaystyle {\ vec {H}} = \ mu ^ {- 1} {\ vec {B}}.}$${\ displaystyle {\ vec {H}}}$${\ displaystyle {\ vec {B}}}$

If a dipole does not point in the direction of an external field, a torque acts on it :

${\ displaystyle {\ vec {M}} = {\ vec {p}} \ times {\ vec {E}}}$ or.
${\ displaystyle {\ vec {M}} = {\ vec {m}} \ times {\ vec {B}} \ Leftrightarrow {\ vec {M}} = {\ vec {m}} _ {H} \ times {\ vec {H}}.}$

If there are two dipoles in the other's field, dipole-dipole forces arise , which decrease with the field gradients . ${\ displaystyle 1 / {r ^ {4}}}$