# 5.3 Maxwell’s Magnetostatic Equations

Thus far, we have introduced the Biot–Savart law for finding the magnetic flux density B and field H due to any distribution of electric currents in free space, and we examined how magnetic fields can exert magnetic forces on moving charged particles and current-carrying conductors. We now examine two additional important properties of magnetostatic fields.

# 5.3.1 Gauss’s Law for Magnetism

In Chapter 4, we learned that the net outward flux of the electric flux density D through a closed surface equals the enclosed net charge Q. We referred to this property as Gauss’s law (for electricity) and expressed it mathematically in differential and integral forms as

∇⋅D=ρ v ⇔ ∮S D⋅d s=Q.(5.43)

Conversion from differential to integral form was accomplished by applying the divergence theorem to a volume υ that is enclosed by a surface S and contains a total charge Q=∫υ ρ v d υ (Section 4-4).

The magnetostatic counterpart of Eq. (5.43), often called Gauss’s law for magnetism, is

The differential form is one of Maxwell’s four equations, and the integral form is obtained with the help of the divergence theorem. Note that the right-hand side of Gauss’s law for magnetism is zero, reflecting the fact that the magnetic equivalence of an electric point charge does not exist in nature.

No matter how many times a permanent magnet is subdivided, each new piece will always have a north and a south pole, even if the process were to be continued down to the atomic level. Consequently, there is no magnetic equivalence to an electric charge q or charge density ρ v.

Formally, the name “Gauss’s law” refers to the electric case, even when no specific reference to electricity is indicated. The property described by Eq. (5.44) has been called “the law of nonexistence of isolated monopoles,” “the law of conservation of magnetic flux,” and “Gauss’s law for magnetism,” among others. We prefer the last of the three cited names because it reminds us of the parallelism, as well as the differences, between the electric and magnetic laws of nature.

The difference between Gauss’s law for electricity and its magnetic counterpart can be elucidated in terms of field lines. Electric field lines originate from positive electric charges and terminate on negative ones. Hence, for the electric field lines of the electric dipole shown in Fig. 5-15(a), the electric flux through a closed surface surrounding one of the charges is nonzero. In contrast, magnetic field lines always form continuous closed loops. As we saw in Section 5-2, the magnetic field lines due to currents do not begin or end at any point; this is true for the linear conductor of Fig. 5-11 and the circular loop of Fig. 5-12, as well as for any current distribution. It is also true for a bar magnet (Fig. 5-15(b)). Because the magnetic field lines form closed loops, the net magnetic flux through any closed surface surrounding the south pole of the magnet (or through any other closed surface) is always zero, regardless of its shape.

# Figure 5-15 Whereas (a) the net electric flux through a closed surface surrounding a charge is not zero, and (b) the net magnetic flux through a closed surface surrounding one of the poles of a magnet is zero.

Figure 5 15 Full Alternative Text

# 5.3.2 Ampère’s Law

In Chapter 4 we learned that the electrostatic field is conservative, meaning that its line integral along a closed contour always vanishes. This property of the electrostatic field was expressed in differential and integral forms as

∇×E=0 ⇔ ∮C E⋅d​ℓ=0.(5.45)

Conversion of the differential to integral form was accomplished by applying Stokes’s theorem to a surface S with contour C.

The magnetostatic counterpart of Eq. (5.45), known as Ampère’s law, is

∇×H=J ⇔ ∮C H⋅d​ℓ=I, (5.46)

where I is the total current passing through S. The differential form again is one of Maxwell’s equations, and the integral form is obtained by integrating both sides of Eq. (5.46) over an open surface S,

∫S(∇×H)⋅d s=∫S J⋅d s,(5.47)

and then invoking Stokes’s theorem with I=∫J⋅d s.

In words, Ampère’s circuital law states that the line integral of H around a closed path is equal to the current traversing the surface bounded by that path. To apply Ampère’s law, the current must flow through a closed path. By way of illustration, for both configurations shown in Figs. 5-16(a) and (b), the line integral of H is equal to the current I, even though the paths have very different shapes and the magnitude of H is not uniform along the path of the configuration in part (b). By the same token, because path (c) in Fig. 5-16 does not enclose the current I, the line integral of H along it vanishes, even though H is not zero along the path.

# Figure 5-16 Ampère’s law states that the line integral of H around a closed contour C is equal to the current traversing the surface bounded by the contour. This is true for contours (a) and (b), but the line integral of H is zero for the contour in (c) because the current I (denoted by the symbol ⊙) is not enclosed by the contour C.

When we examined Gauss’s law in Section 4-4, we discovered that in practice its usefulness for calculating the electric flux density D is limited to charge distributions that possess a certain degree of symmetry and that the calculation procedure is subject to the proper choice of a Gaussian surface enclosing the charges. A similar restriction applies to Ampère’s law: its usefulness is limited to symmetric current distributions that allow the choice of convenient Ampèrian contours around them, as illustrated by Examples 5-4 to 5-6.

# Example 5-4: Magnetic Field of a Long Wire

A long (practically infinite) straight wire of radius a carries a steady current I that is uniformly distributed over its cross section. Determine the magnetic field H a distance r from the wire axis for (a) r≤a (inside the wire) and (b) r≥a (outside the wire).

Solution: (a) We choose I to be along the +z direction (Fig. 5-17(a)). To determine H 1=H at a distance r=r 1≤a, we choose the Ampèrian contour C 1 to be a circular path of radius r=r 1 (Fig. 5-17(b)). In this case, Ampère’s law takes the form

∮C 1 H 1⋅d l 1=I 1,(5.48)

where I 1 is the fraction of the total current I flowing through C 1. From symmetry, H 1 must be constant in magnitude and parallel to the contour at any point along the path. Furthermore, to satisfy the right-hand rule and given that I is along the z direction, H 1 must be in the +ϕ direction. Hence, H 1=ϕ ˆ H 1, d l 1=ϕ ˆ r 1 d ϕ, and the left-hand side of Eq. (5.48) becomes

∮C 1 H 1⋅d l 1=∫0 2 π H 1(ϕ ˆ⋅ϕ ˆ)r 1 d ϕ=2 π r 1 H 1.

# Figure 5-17 Infinitely long wire of radius a carrying a uniform current I along the +z direction: (a) general configuration showing contours C 1 and C 2; (b) cross-sectional view; and (c) a plot of H versus r (Example 5-4).

(a) Cylindrical wire

Figure 5 17-a Full Alternative Text

(b) Wire cross section

Figure 5 17-b Full Alternative Text
Figure 5 17-c Full Alternative Text

The current I 1 flowing through the area enclosed by C 1 is equal to the total current I multiplied by the ratio of the area enclosed by C 1 to the total cross-sectional area of the wire:

I 1=π r 1 2 π a 2 I=r 1 a 2 I.

Equating both sides of Eq. (5.48) and then solving for H 1 yields

H 1=ϕ ˆ H 1=ϕ ˆ r 1 2 π a 2 I (for r 1≤a).(5.49a)

(b) For r=r 2≥a, we choose path C 2, which encloses all the current I. Hence, H 2=ϕ ˆ H 2, d​ℓ 2=ϕ ˆ r 2 d ϕ, and

∮ C 2 H 2 ⋅d l 2 =2 π r 2 H 2 =I,

which yields

H 2=ϕ ˆ H 2=ϕ ˆ I 2 π r 2 (for r 2≥a).(5.49b)

Ignoring the subscript 2, we observe that Eq. (5.49b) provides the same expression for B=μ 0 H as Eq. (5.30), which was derived on the basis of the Biot–Savart law.

The variation of the magnitude of H as a function of r is plotted in Fig. 5-17(c); H increases linearly between r=0 and r=a (inside the conductor), and then decreases as 1/r for r>a (outside the conductor).

# Example 5-5: Magnetic Field inside a Toroidal Coil

A toroidal coil (also called a torus or toroid) is a doughnut-shaped structure (called the core) wrapped in closely spaced turns of wire (Fig. 5-18). For clarity, we show the turns in the figure as spaced far apart, but in practice, they are wound in a closely spaced arrangement to form approximately circular loops. The toroid is used to magnetically couple multiple circuits and to measure the magnetic properties of materials, as illustrated later in Fig. 5-31. For a toroid with N turns carrying a current I, determine the magnetic field H in each of the following three regions: r<a, a<r<b, and r>b with all in the azimuthal symmetry plane of the toroid.

# Figure 5-18 Toroidal coil with inner radius a and outer radius b. The wire loops usually are much more closely spaced than shown in the figure (Example 5-5).

Figure 5 18 Full Alternative Text

Solution: From symmetry, it is clear that H is uniform in the azimuthal direction. If we construct a circular Ampèrian contour with center at the origin and radius r<a, there will be no current flowing through the surface of the contour. Therefore,

H=0 for r<a.

Similarly, for an Ampèrian contour with radius r>b, the net current flowing through its surface is zero because an equal number of current coils cross the surface in both directions; hence,

H=0 for r>b (region exterior to the toroidal coil).

For the region inside the core, we construct a path of radius r (Fig. 5-18). For each loop, we know from Example 5-3 that the field H at the center of the loop points along the axis of the loop, which in this case is the ϕ direction, and in view of the direction of the current I shown in Fig. 5-18, the right-hand rule tells us that H must be in the −ϕ direction. Hence, H=−ϕ ˆ H. The total current crossing the surface of the contour with radius r is N I and its direction is into the page. According to the right-hand rule associated with Ampère’s law, the current is positive if it crosses the surface of the contour in the direction of the four fingers of the right hand when the thumb is pointing along the direction of the contour C. Hence, the current through the surface spanned by the contour is −N I. Application of Ampère’s law then gives

∮C H⋅d l=∫0 2 π(−ϕ ˆ H)⋅ϕ ˆ r d ϕ=−2 π r H=−N I.

Hence, H=N I/(2 π r) and

H=−ϕ ˆ H=−ϕ ˆ N I 2 π r (for a<r<b).(5.50)

# Example 5-6: Magnetic Field of an Infinite Current Sheet

The x−y plane contains an infinite current sheet with surface current density J s=x ˆ J s (Fig. 5-19). Find the magnetic field H everywhere in space.

# Figure 5-19 A thin current sheet in the x−y plane carrying a surface current density J s=x ˆ J s (Example 5-6).

Figure 5 19 Full Alternative Text

Solution: From symmetry considerations and the right-hand rule, for z>0 and z<0 H must be in the directions shown in the figure. That is,

H={ −y ˆ H for z>0, y ˆ H for z<0.

To evaluate the line integral in Ampère’s law, we choose a rectangular Ampèrian path around the sheet with dimensions l and w (Fig. 5-19). Recalling that J s represents current per unit length along the y direction, the total current crossing the surface of the rectangular loop is I=J s l. Hence, applying Ampère’s law over the loop in a counterclockwise direction, while noting that H is perpendicular to the paths of length w, we have

∮C H⋅d l=2 H l=J s l,

from which we obtain the result

H={ −y ˆ J s 2 for z>0, y ˆ J s 2 for z<0. (5.51)

Concept Question 5-8: What are the fundamental differences between electric and magnetic fields?

Concept Question 5-9: If the line integral of H over a closed contour is zero, does it follow that H=0 at every point on the contour? If not, what then does it imply?

Concept Question 5-10: Compare the utility of applying the Biot–Savart law versus applying Ampère’s law for computing the magnetic field due to current-carrying conductors.

Concept Question 5-11: What is a toroid? What is the magnetic field outside the toroid?