# 5.4 Vector Magnetic Potential

In our treatment of electrostatic fields in Chapter 4, we defined the electrostatic potential V as the line integral of the electric field E and found that V and E are related by E=−∇V. This relationship proved useful not only in relating electric field distributions in circuit elements (such as resistors and capacitors) to the voltages across them but also to determine E for a given charge distribution by first computing V using Eq. (4.48). We now explore a similar approach in connection with the magnetic flux density B.

According to Eq. (5.44), ∇⋅B=0. We wish to define B in terms of a magnetic potential with the constraint that such a definition guarantees that the divergence of B is always zero. This can be realized by taking advantage of the vector identity given by Eq. (3.106b), which states that, for any vector A,

∇⋅(∇×A)=0.(5.52)

Hence, by introducing the vector magnetic potential A such that

we are guaranteed that ∇⋅B=0. The SI unit for B is the tesla (T). An equivalent unit is webers per square meter (Wb/m 2). Consequently, the SI unit for A is (Wb/m).

The train basically floats at a height of 1 or more centimeters above the track, which is made possible by magnetic levitation (Fig. TF10-5(b)). The train carries superconducting electromagnets that induce currents in coils built into the guide rails alongside the train. The magnetic interaction between the train’s superconducting electromagnets and the guide-rail coils serves not only to levitate the train but also to propel it along the track.

# 5.4.1 Vector Poisson’s Equation

With B=μ H, the differential form of Ampère’s law given by Eq. (5.46) can be written as

∇×B=μ J.(5.54)

If we substitute Eq. (5.53) into Eq. (5.54), we obtain

∇×(∇×A)=μ J.(5.55)

For any vector A, the Laplacian of A obeys the vector identity given by Eq. (3.113), that is,

∇2 A=∇(∇⋅A)−∇×(∇×A),(5.56)

where, by definition, ∇2 A in Cartesian coordinates is

∇2 A=(∂2∂x 2+∂2∂y 2+∂2∂z 2)A=x ˆ∇2 A x+y ˆ∇2 A y+z ˆ∇2 A z.(5.57)

Combining Eq. (5.55) with Eq. (5.56) gives

∇(∇⋅A)−∇2 A=μ J.(5.58)

This equation contains a term involving ∇⋅A. It turns out that we have a fair amount of latitude in specifying a value or a mathematical form for ∇⋅A without conflicting with the requirement represented by Eq. (5.53). The simplest among these allowed restrictions on A is

∇⋅A=0.(5.59)

Using this choice in Eq. (5.58) leads to the vector Poisson’s equation

Using the definition for ∇2 A given by Eq. (5.57), the vector Poisson’s equation can be decomposed into three scalar Poisson’s equations:

In electrostatics, Poisson’s equation for the scalar potential V is given by Eq. (4.70) as

∇2 V=−ρ v ε,(5.62)

and its solution for a volume charge distribution ρ v occupying a volume υ′ was given by Eq. (4.71) as

V=1 4 π ε∫υ′ρ v R′ d υ′.(5.63)

Poisson’s equations for A x, A y, and A z are mathematically identical in form to Eq. (5.62). Hence, for a current density J with x component J x distributed over a volume υ′, the solution for Eq. (5.61a) is

A x=μ 4 π∫υ′J x R′d υ′ (Wb/m).(5.64)

Similar solutions can be written for A y in terms of J y and for A z in terms of J z. The three solutions can be combined into a vector equation:

In view of Eq. (5.23), if the current distribution is specified over a surface S′, then J d υ′ should be replaced with J s d s′ and υ′ should be replaced with S′; similarly, for a line distribution, J d υ′ should be replaced with I d l′, and the integration should be performed over the associated path l′.

For a specified current distribution, Eq. (5.65) can be used to find A, and then Eq. (5.53) can be used to find B. Except for simple current distributions with symmetrical geometries that lend themselves to the application of Ampère’s law, in practice we often use the approaches provided by the Biot–Savart law and the vector magnetic potential, and among these two, the latter often is more convenient to apply because it is easier to perform the integration in Eq. (5.65) than that in Eq. (5.22).

# 5.4.2 Magnetic Flux

The magnetic flux Φ linking a surface S is defined as the total magnetic flux density passing through it, or

Φ=∫S B⋅d s (Wb).(5.66)

If we insert Eq. (5.53) into Eq. (5.66) and then invoke Stokes’s theorem, we obtain

Φ=∫S(∇×A)⋅d s=∮C A⋅d l (Wb),(5.67)

where C is the contour bounding the surface S. Thus, Φ can be determined by either Eq. (5.66) or Eq. (5.67), whichever is easier to integrate for the specific problem under consideration.