Frozen-in integral
The integral of the induction equation of a magnetic field for the limiting case of an ideally-conducting medium. The physical meaning of the frozen-in integral is that by moving the liquid, the force lines of the magnetic field are moved together with the liquid particles.
For the motion of an ideally-conducting medium the magnetic field strength $ \mathbf H $ is described by the equation:
$$
\frac{d}{dt }
\left ( { \frac{\mathbf H} \rho
} \right ) = \
\left ( { \frac{\mathbf H} \rho
} , \nabla \right ) \mathbf v ,
$$
where $ \rho $ is the density and $ \mathbf v $ is the rate of motion of the medium. A change in the line element $ d \mathbf l $ of a force line of the magnetic field is described by the equation
$$
\frac{d}{dt }
d \mathbf l = ( d \mathbf l , \nabla ) \mathbf v .
$$
The vectors $ \mathbf H $ and $ d \mathbf l $ are collinear:
$$ { \frac{\mathbf H} \rho
} = \textrm{ const } \cdot d \mathbf l .
$$
The following equation, which goes by the name of frozen-in integral, is valid:
$$
\frac{\mathbf H d \mathbf l _ {0} } \rho
= \
\frac{\mathbf H _ {0} }{\rho _ {0} }
d \mathbf l ,
$$
where the index "0" refers to the values of the parameters at the initial moment of time.
It follows from the frozen-in integral that the magnetic flux of a field across any surface, encircled by a contour of liquid particles, is independent of time.
References
| [1] | T.G. Cowling, "Magneto-hydrodynamics" , Interscience (1957) |
| [2] | L.D. Landau, E.M. Lifshitz, "Electrodynamics of continous media" , Pergamon (1960) (Translated from Russian) |
| [3] | A.G. Kulikovskii, G.A. Lyubimov, "Magnetic hydrodynamics" , Moscow (1962) (In Russian) |
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