Irodov 3.275: Molecular Currents That Cancel Outside

Problem Irodov 3.275

A long, straight, uniform wire of circular cross-section is made of a paramagnetic material of susceptibility $\chi$ and carries a steady current $I$ . Find:

the molecular (magnetization) current $I'_s$ flowing on the wire’s surface;
the molecular current $I'_v$ flowing through its volume. How are the two currents directed relative to each other?

A piece of the long paramagnetic wire (susceptibility χ) carrying the current I along its axis. — A piece of the long paramagnetic wire (susceptibility $\chi$ ) carrying the current $I$ along its axis.

Given

steady current $I$
susceptibility $\chi$
long round uniform wire

Find

(a)

I'_s

; (b)

I'_v

and their relative directions

Solution

Step 1H inside the wire

Let the wire have radius $a$ . By symmetry $\vect{H}$ circles the axis and has the same magnitude all round a circle of radius $r$ , so its circulation round that circle equals the free current inside it, which for a uniform current density is the fraction $r^2/a^2$ of $I$ :

H\cdot 2\pi r = I\,\frac{r^2}{a^2}

so that inside

H = \frac{Ir}{2\pi a^2} \tag{1}

Its sense round the axis is the right-hand rule about $I$ .

Step 2The magnetization

In a paramagnet the magnetization is $\vect{J} = \chi\vect{H}$ , so it too circles the axis, in the same sense as $\vect{H}$ , with magnitude from (1)

J = \frac{\chi I r}{2\pi a^2} \tag{2}

It grows from zero on the axis to its largest value at the surface, $J(a) = \chi I/2\pi a$ .

Step 3(a) The surface current

A magnetization that ends at a surface leaves a molecular current on it of linear density $\vect{J}\times\vect{n}$ , $\vect{n}$ the outward normal. Here $\vect{J}$ is tangential round the wire and $\vect{n}$ radial, so the current runs along the wire, with density $J(a)$ , and in the sense of $\hat{\vect{\varphi}}\times\hat{\vect{r}} = -\hat{\vect{z}}$ when $I$ flows along $+\hat{\vect{z}}$ : against $I$ . Over the whole circumference, using (2) at $r = a$ ,

|I'_s| = J(a)\cdot 2\pi a = \frac{\chi I}{2\pi a}\cdot 2\pi a

I'_s = \chi I

(3)

It runs against $I$ .

Step 4(b) The volume current

Inside, the molecular current density is the curl of the magnetization, and since $\vect{J} = \chi\vect{H}$ with $\chi$ constant and $\nabla\times\vect{H} = \vect{j}$ (the conduction current density),

\vect{j}' = \nabla\times\vect{J} = \chi\,\nabla\times\vect{H} = \chi\,\vect{j}

The volume molecular current is uniform, along the conduction current, and $\chi$ times as dense. Integrated over the cross-section,

I'_v = \chi I

(4)

The two molecular currents are equal in magnitude and opposite in direction: the volume current flows with $I$ through the body of the wire, the surface current returns against it in the skin.

The wire in axial section: the magnetization J circles the axis (out of the page above it, into the page below), the volume molecular current I'_v runs with I inside, and the surface molecular current I'_s runs back along the surface. — The wire in axial section: the magnetization $\vect{J}$ circles the axis (out of the page above it, into the page below), the volume molecular current $I'_v$ runs with $I$ inside, and the surface molecular current $I'_s$ runs back along the surface.

Check

The circulation of $\vect{J}$ round a circle just inside the surface equals the molecular current threading it, which is only the volume part: $J(a)\cdot 2\pi a = \chi I = I'_v$ , as found. Round a circle just outside, where $\vect{J} = 0$ , the threading molecular current must be zero, and indeed the signed sum of the two, counted along $I$ , is $\chi I + (-\chi I) = 0$ . With $\chi = 0$ there is no magnetization and both currents vanish.

Remark

Because the two molecular currents cancel, the field outside the wire is exactly the vacuum value $\mu_0 I/2\pi r$ , whatever $\chi$ is; only inside is $B$ raised, by the factor $\mu = 1 + \chi$ , to $\mu\mu_0 Ir/2\pi a^2$ . The same cancellation holds for any magnetized body: the total molecular current crossing any complete cross-section is zero, because the magnetization circulates in closed loops and never transports charge along the wire.