Current-Carrying Wire in Uniform Magnetic Field
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Consider an infinitely long cylindrical wire that carries a current in the perpendicular uniform magnetic field . This Demonstration shows the resultant field in the spaces both inside and outside the cylinder. Assume that the current density is constant in the cross section of wire and that the relative permeabilities , inside and outside the cylinder are different. The directions of the applied field and wire current are the and axes, respectively. The field pattern is plotted in the - plane using color for strength and arrows for direction; the field intensity is normalized by . You can vary the relative permeabilities, the applied magnetic field , the wire diameter , and the wire current .
Contributed by: Y. Shibuya (October 2013)
Open content licensed under CC BY-NC-SA
Snapshot 1: the 10mm-diameter nonmagnetic wire carries (i.e. ) in field
Snapshot 2: the 10mm-diameter nonmagnetic wire carries (i.e. ) in field
Snapshot 3: the 10mm-diameter ferromagnetic cylinder of is in field
The resultant field both inside and outside the cylinder can be obtained considering the vector potentials of the uniform field and the wire current . As the result, the field is described in the following equations in cylindrical coordinates :
Here is the radius of the cylinder (or , using a wire of diameter ). The third component is zero. Those fields are transformed to Cartesian in the graphics displayed.
For Snapshots 1 and 2, the magnetic field at the wire surface due to the wire current is calculated to be and , respectively. Since in those examples, they correspond to the cases and . The unsymmetrical field patterns suggest that an upward electromagnetic force should be generated.
Snapshot 3 shows the effect of an outside magnetic field on the ferromagnetic material. According to the above equation, the field inside the cylinder is , with direction along the axis, the direction of the applied field. The field intensity is very small since .
 J. A. Stratton, Electromagnetic Theory, New York: McGraw–Hill, 1941.