# Analysis of electromagnetic pressure on thin plate melt

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## Abstract

An equation used to calculate electromagnetic pressure for confining and shaping a plate-form melt with small or medium thickness is brought forward. The equation gives a precise relationship between electromagnetic pressure factor*p* and current frequency*f*, melt thickness a and electromagnetic parameter μγ. The main results can be summarized as follows: (i) As the*f, a* and μγ are large enough, causing the ratio of melt thickness to current skin depth*a/ δ* to be larger than 2.2,*p* will be a constant (*p* = 1) and the electromagnetic pressure can be calculated with*P* _{m} =*B* ^{2}/2μ. (ii) As the*f, a*, and μγ are not large enough, and the ratio*a/ δ* is less than 2.2,*p* is no longer a constant but changes from 1 to 0 with the decrease of*a/δ*. In this case, the value of electromagnetic pressure is determined not only by magnetic flux density*B* but also by the current frequency*f*, melt thickness a and electromagnetic parameter μγ.

## Keywords

electromagnetic confinement and shaping electromagnetic pressure magnetic flux density## Symbols

- P
_{m} Electromagnetic pressure

- p.
electromagnetic pressure factor

- P.
static pressure by melt

- F
the value of electromagnetic body force in melt

- B
the value of magnetic flux density on melt surface

- B
_{2} the value of magnetic flux density in melt

- B
_{zo} the value of magnetic flux density at the center (

*x*=0) of melt- (B
_{z0})_{m} amplitude of magnetic flux density at the center (

*x*=0) of melt- B
_{z} the vector of magnetic flux density in melt

- H
_{z} the vector of magnetic field density in melt

- E
_{2} the vector of electric field density in melt

- J
_{1} the vector of electric current density in melt

- B
_{z} the phase form of magnetic flux density in melt

- B
_{z0} the phase form of magnetic flux density at the center (

*x*= 0) of melt- H
_{z} the phase form of magnetic field density in melt

- Ė
_{0} the phase form of electric field density in melt

- J.
the phase form of electric current density in melt

- j
the imaginary operator of complex number

- μ
magnetic conductivity of melt

- γ
electric conductivity of melt

- α
melt thickness

- f
electric current frequency

- δ
electric current theoretic skin depth

- K
√ηfμγ

- k
√2jK

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## References

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