Frozen-in Theorem

Intuition

In a perfectly conducting fluid, magnetic field lines cannot end; they must move together with the plasma. Imagine drawing field lines on a rubber sheet and stretching the sheet—the lines go wherever the material goes. This is the frozen-in property of ideal MHD.

Formal Definition

Let $\mathbf{B}/\rho$ be the magnetic flux per unit mass. In an ideal, incompressible conductor:

\[\frac{D}{Dt}\left(\frac{\mathbf{B}}{\rho}\right) = 0\]

For incompressible flow ($\rho=$ const), the field lines are simply advected:

\[\frac{\partial \mathbf{B}}{\partial t} = \nabla\times(\mathbf{U}\times\mathbf{B})\]

Mathematical Formulation

Start with ideal Ohm’s law:

\[\mathbf{E} + \mathbf{U}\times\mathbf{B} = 0\]

Combine with Faraday’s law $\partial \mathbf{B}/\partial t = -\nabla\times\mathbf{E}$:

\[\frac{\partial \mathbf{B}}{\partial t} = \nabla\times(\mathbf{U}\times\mathbf{B})\]

For an incompressible fluid, this implies that magnetic flux $\oint \mathbf{B}\cdot d\mathbf{S}$ through any fluid element is conserved in time.

Derivation

Write Faraday’s law: $\partial \mathbf{B}/\partial t = -\nabla\times\mathbf{E}$.
Insert ideal Ohm’s law: $\mathbf{E} = -\mathbf{U}\times\mathbf{B}$.
Use vector identity: $\nabla\times(\mathbf{U}\times\mathbf{B}) = (\mathbf{B}\cdot\nabla)\mathbf{U} - (\mathbf{U}\cdot\nabla)\mathbf{B} - \mathbf{B}(\nabla\cdot\mathbf{U})
\mathbf{U}(\nabla\cdot\mathbf{B})$.
With $\nabla\cdot\mathbf{B}=0$, this becomes the induction equation.
Interpret $\partial \mathbf{B}/\partial t$ as the change of $\mathbf{B}$ seen by an observer moving with velocity $\mathbf{U}$.

Worked Example

Solar corona: A coronal loop is a bundle of field lines “frozen” into the plasma. If photospheric motions twist the footpoints, the loop twists like a rope. Reconnection can release the magnetic energy in a flare.

Mathematically: if $B_\phi(r)$ grows with twist, the magnetic helicity $K = \int \mathbf{A}\cdot\mathbf{B}\, dV$ is conserved until a non-ideal event breaks the topology.

Common Mistakes

Frozen-in is exact at all scales. No; reconnection breaks it when $\eta \neq 0$ or at kinetic scales where electron inertia matters.
Field lines are material strings. They are topological constructs; they do not have a finite “thickness” or rest mass.
Incompressibility is required. The frozen-in theorem holds more generally as flux conservation even in compressible flows.

Quiz Questions

Conceptual: Explain in one sentence why an perfect conductor cannot allow magnetic flux to pass through its interior if it was initially excluded.
Computational: Show that $\nabla\cdot\mathbf{B}=0$ is preserved by the induction equation.
MCQ: The frozen-in property fails when:
A) $\eta = 0$
B) $\eta \neq 0$ and $\nabla\times\mathbf{E} \neq -\mathbf{U}\times\mathbf{B}$
C) The plasma is collisionless
D) $\mathbf{U} = 0$