## Force and Heat Generation in a Conducting Sphere in an Alternating Magnetic Field

##### Author

Sathuvalli, Udaya Bhaskar R.

##### Date

1995##### Advisor

Bayazitoglu, Yildiz

##### Degree

Doctor of Philosophy

##### Abstract

The interaction of an electrically conducting sphere with a time varying magnetic ﬁeld is useful in the study of "containerless" processing methods such as electromagnetic levitation melting. The fundamental quantities of interest in this interaction are the rate of heat generation in the sphere, the Lorentz force and magnetic pressure on it. These quantities depend upon the nature of the current sources that create the magnetic field, and the material properties of the sphere. In this work, the Maxwell equations for the interaction of a sphere with an arbitrary external alternating magnetic ﬁeld are ﬁrst formulated. Then, the density of the induced currents in the sphere is found as a function of the external current sources and the material properties of the sphere. The current density is now used to calculate the heat generated in the sphere. Next, a method to calculate the Lorentz force on an electrically conducting sphere placed in an arbitrary sinusoidally varying magnetic ﬁeld is developed and a formula for the force on the sphere is given. This formula is used to derive the special case of a sphere in an axisymmetric system of circular current loops. Numerical results for the force on a sphere on the axis of a stack of loops are presented as a function of the stack geometry. The results for the heat generation and the Lorentz force obtained in this study are compared with the results obtained by a previously used model (known as the "homogeneous model") which assumes that the external magnetic ﬁeld is uniform and unidirectional. It is shown that the homogeneous model is a special case of the present model and that it underestimates heat generation signiﬁcantly, and overestimates the Lorentz force. In addition, as the size of the sphere decreases, the homogeneous model gives erroneous results, approaching an order of magnitude for heat generation in a very small sphere. Subsequently, a procedure to determine the magnetic pressure distribution on the surface of a levitated liquid metal droplet is developed. The pressure distribution is calculated in terms of the geometry of the coil that creates the ﬁeld. Finally, the magnetic ﬁelds of helical windings that are commonly used in the laboratory for levitation melting are calculated.

##### Keyword

Mechanical engineering; Materials science