Skip to content

The Magnetic Stress Tensor

Source lecture(s): PC368 Lec 14

Intuition

Faraday's mental picture of field lines as elastic bands under tension, pushing each other apart sideways, is not a metaphor — it is an exact rewriting of the \(\mathbf{J}\times\mathbf{B}\) force. Once you see it, most of MHD equilibrium and stability becomes mechanical intuition.

The decomposition

Using Ampère's law \(\mu_0\mathbf{J} = \nabla\times\mathbf{B}\) and a vector identity:

\[\mathbf{J}\times\mathbf{B} = \underbrace{-\nabla\left(\frac{B^2}{2\mu_0}\right)}_{\text{magnetic pressure}} + \underbrace{\frac{(\mathbf{B}\cdot\nabla)\mathbf{B}}{\mu_0}}_{\text{magnetic tension}}\]

equivalently \(\mathbf{J}\times\mathbf{B} = \nabla\cdot\overleftrightarrow{T}\) with

\[T_{ij} = \frac{1}{\mu_0}\left(B_iB_j - \frac{B^2}{2}\delta_{ij}\right)\]

Read the tensor in a frame with \(\hat{z}\) along \(\mathbf{B}\): it is \(\mathrm{diag}(-p_B,\,-p_B,\,+p_B)\) with \(p_B = B^2/2\mu_0\). Pressure \(B^2/2\mu_0\) across the field, tension \(B^2/\mu_0\) along it. The tension term is only non-zero when field lines are curved: writing \((\mathbf{B}\cdot\nabla)\mathbf{B}/\mu_0 = -\dfrac{B^2}{\mu_0 R_c}\hat{\mathbf{n}}\), the tension pulls toward the centre of curvature with force per unit volume \(B^2/\mu_0R_c\).

Plasma beta

Equilibrium balances plasma pressure against magnetic stress, so their ratio is the figure of merit:

\[\beta = \frac{p}{B^2/2\mu_0} = \frac{2\mu_0 nk_BT}{B^2}\]
  • \(\beta \ll 1\): magnetically dominated. Field lines dictate; plasma follows. The solar corona (\(\beta\sim10^{-2}\)), tokamak edge.
  • \(\beta \gg 1\): plasma dominated. Flows drag the field around. The solar interior, the heliosphere far from the Sun.
  • \(\beta \sim 1\): the interesting, unstable middle — and the economically necessary regime for a reactor, since fusion power goes as \(p^2\) while magnet cost goes as \(B^2\). A reactor's profitability is essentially \(\beta\).

What it explains immediately

  • The Z-pinch. Azimuthal field around an axial current is curved, so tension squeezes inward — pinch equilibria in one sentence.
  • Why a purely toroidal field cannot confine. Tension from curvature points toward the major axis, and there is nothing balancing the hoop force outward.
  • Alfvén waves. Tension is a restoring force on a string of mass density \(\rho\), so it propagates transverse waves at \(v_A = B/\sqrt{\mu_0\rho}\) — literally the string formula \(\sqrt{\text{tension}/\text{mass per length}}\).
  • Reconnection outflow. Newly reconnected field lines are sharply bent; tension slingshots the plasma out of the X-point at \(v_A\).
  • Solar prominences. Cool dense plasma sits for weeks in magnetic dips, held against gravity by tension.

Common mistakes

  • Adding magnetic pressure as if it were isotropic. It is not — the tensor has opposite signs along and across \(\mathbf{B}\). Total pressure \(p + B^2/2\mu_0\) is correct only across the field.
  • Forgetting tension needs curvature. In a straight uniform field the tension term vanishes identically; only pressure acts.
  • Believing \(\mathbf{J}\times\mathbf{B}\) always confines. It has no fixed sign. The virial theorem proves that no configuration can confine itself with its own fields alone.

Knowledge graph position

Prerequisites: MHD, momentum equation. Leads to: MHD equilibrium, pinch equilibria, energy principle.

Quiz

Q1 (conceptual). Why is magnetic tension zero in a straight uniform field?

Answer

The tension term is \((\mathbf{B}\cdot\nabla)\mathbf{B}/\mu_0\), which measures how \(\mathbf{B}\) changes along itself. Straight and uniform means no change, so no tension force — only pressure, and a uniform pressure exerts no net force either.

Q2 (computational). A tokamak has \(B = 5\) T, \(n = 10^{20}\) m⁻³, \(T = 10\) keV (both species). What is \(\beta\)?

Answer

\(p = 2nk_BT = 2\times10^{20}\times10^4\times1.6\times10^{-19} = 3.2\times10^5\) Pa. \(B^2/2\mu_0 = 25/(2\times4\pi\times10^{-7}) = 9.9\times10^6\) Pa. \(\beta = 3.2\times10^5/9.9\times10^6 \approx 0.032\) — about 3%, typical for a conventional tokamak and a large part of why they must be big.

Q3 (MCQ). Reconnection outflow reaches the Alfvén speed because:

  • (a) magnetic pressure pushes the plasma out
  • (b) tension in the sharply bent, newly reconnected field lines slingshots the plasma
  • (c) thermal pressure in the current sheet is very high
  • (d) the electric field accelerates the ions
Answer

(b). Field lines leaving the X-point are strongly curved, and tension \(B^2/\mu_0R_c\) straightens them, flinging the frozen-in plasma along. The natural speed for a tension-driven process in a medium of density \(\rho\) is \(v_A\).