## Magnetic potential |

Part of a series of articles about |

The term **magnetic potential** can be used for either of two quantities in *magnetic vector potential*, or simply *vector potential*, **A**; and the *magnetic scalar potential* *ψ*. Both quantities can be used in certain circumstances to calculate the **B**.

The more frequently used magnetic vector potential is defined so that its **A** = **B**. Together with the *φ*, the magnetic vector potential can be used to specify the **E** as well. Therefore, many equations of electromagnetism can be written either in terms of the fields **E** and **B**, or equivalently in terms of the potentials **A** and *φ*. In more advanced theories such as

The magnetic scalar potential *ψ* is sometimes used to specify the magnetic **H**-field*ψ* is to determine the magnetic field due to ^{[citation needed]}

Historically, ^{[1]}

- magnetic vector potential
- magnetic scalar potential
- see also
- notes
- references

The magnetic vector potential **A** is a *ϕ* (a ^{[2]}

where **B** is the **E** is the *vector potential* and *scalar potential* are used for *magnetic vector potential* and * electric potential*, respectively. In mathematics,

If electric and magnetic fields are defined as above from potentials, they automatically satisfy two of **A** is continuous and well-defined everywhere, then it is guaranteed not to result in **A** is allowed to be either undefined or multiple-valued in some places; see

Starting with the above definitions:

Alternatively, the existence of **A** and *ϕ* is guaranteed from these two laws using **∇** ⋅ **B** = 0), **A** always exists that satisfies the above definition.

The vector potential **A** is used when studying the

In the **A** are ^{−1} and are the same as that of

Although the magnetic field **B** is a **A** is a ^{[3]} This means that if the **B** would switch signs, but **A** would not change. This is an example of a general theorem: The curl of a polar vector is a pseudovector, and vice versa.^{[3]}

The above definition does not define the magnetic vector potential uniquely because, by definition, we can arbitrarily add **A**. This condition is known as

Using the above definition of the potentials and applying it to the other two Maxwell's equations (the ones that are not automatically satisfied) results in a complicated differential equation that can be simplified using the **A** is chosen to satisfy:

^{[2]}

Using the Lorenz gauge, **A** and the *ϕ*:^{[2]}

In other

The solutions of Maxwell's equations in the Lorenz gauge (see Feynman ^{[2]} and Jackson^{[4]}) with the boundary condition that both potentials go to zero sufficiently fast as they approach infinity are called the **A**(**r**, *t*) and the electric scalar potential *ϕ*(**r**, *t*) due to a current distribution of **J**(**r**′, *t*′), *ρ*(**r**′, *t*′), and *ρ* and **J** are non-zero at least sometimes and some places):

where the fields at **r** and time *t* are calculated from sources at distant position **r**′ at an earlier time *t*′. The location **r**′ is a source point in the charge or current distribution (also the integration variable, within volume Ω). The earlier time *t*′ is called the * retarded time*, and calculated as

- .

There are a few notable things about **A** and *ϕ* calculated in this way:

- (The
Lorenz gauge condition ): is satisfied. - The position of
**r**, the point at which values for*ϕ*and**A**are found, only enters the equation as part of the scalar distance from**r**′ to**r**. The direction from**r**′ to**r**does not enter into the equation. The only thing that matters about a source point is how far away it is. - The integrand uses
,retarded time *t*′. This simply reflects the fact that changes in the sources propagate at the speed of light. Hence the charge and current densities affecting the electric and magnetic potential at**r**and*t*, from remote location**r**′ must also be at some prior time*t*′. - The equation for
**A**is a vector equation. In Cartesian coordinates, the equation separates into three scalar equations:^{[5]}

- In this form it is easy to see that the component of
**A**in a given direction depends only on the components of**J**that are in the same direction. If the current is carried in a long straight wire, the**A**points in the same direction as the wire.

In other gauges, the formula for **A** and *ϕ* is different; for example, see

See Feynman^{[6]} for the depiction of the **A** field around a long thin

Since

assuming quasi-static conditions, i.e.

the lines and contours of **A** relate to **B** like the lines and contours of **B** relate to **j**. Thus, a depiction of the **A** field around a loop of **B** flux (as would be produced in a **B** field around a loop of current.

The figure to the right is an artist's depiction of the **A** field. The thicker lines indicate paths of higher average intensity (shorter paths have higher intensity so that the path integral is the same). The lines are drawn to (aesthetically) impart the general look of the **A**-field.

The drawing tacitly assumes **∇** ⋅ **A** = 0, true under one of the following assumptions:

- the
Coulomb gauge is assumed - the
Lorenz gauge is assumed and there is no distribution of charge,*ρ*= 0, - the
Lorenz gauge is assumed and zero frequency is assumed - the
Lorenz gauge is assumed and a non-zero frequency that is low enough to neglect is assumed

In the context of *four-potential*.

One motivation for doing so is that the four-potential is a mathematical

Another, related motivation is that the content of classical electromagnetism can be written in a concise and convenient form using the electromagnetic four potential, especially when the

where □ is the *J* is the

Other Languages

català: Potencial escalar magnètic

español: Potencial escalar magnético

فارسی: پتانسیل برداری مغناطیسی

français: Potentiel vecteur du champ magnétique

italiano: Potenziale magnetico

עברית: פוטנציאל וקטורי (פיזיקה)

magyar: Vektorpotenciál (fizika)

मराठी: चुंबकी विभव

Nederlands: Vectorpotentiaal

ਪੰਜਾਬੀ: ਚੁੰਬਕੀ ਪੁਟੈਂਸ਼ਲ

polski: Potencjał magnetyczny

português: Potencial magnético

shqip: Potenciali magnetik

татарча/tatarça: Электромагнит кыры вектор потенциалы

中文: 磁矢势