Fields of Rings and Coils

Fields of Rings and Coils - of length dl Fortunately dl and...

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Fields of Rings and Coils Equipped with our power calculus equation, we can now derive the field created by rings and coils. Field of a Single Ring Consider a single wire wrapped in a circle, and carrying a current. From our second right hand rule, we can describe qualitatively the magnetic field created by the current. Shown below is such a field: Figure %: The field created by a ring. If the ring lies in the x - y plane, then the field lines point in the positive z direction It is clear that on the axis of the ring, the field lines point straight up, perpendicular to the plane of the ring. Notice the similarity between the field of a ring and that of a magnet. This is not a coincidence, and can be described using atomic theory of ferromagnetic materials. We can also determine the strength of this field on the axis. Consider a point on the axis, elevated a distance z from the plane of a ring with radius b , shown below. Figure %: A point of the axis of the ring, shown with relevant distances and angles to an element
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Unformatted text preview: of length, dl . Fortunately, dl and are perpendicular in this case, greatly simplifying our equation for dB : dB = However, this vector is at an angle θ to the z axis. Thus the component of the field produced by dl in the z-axis is given by: dB z = cos θ = The geometry used to get this equation can be seen from the . Now we integrate this expression over the entire circle. Notice, however, that dl = 2 Πb , or simply the circumference of the circle. Thus: B z = = This equation applies to any point on the axis of the ring. To find the field at the center of the ring, we simply plug in z = 0 : B z = Thus we have a set of equations for the field of a ring. Though the derivation required calculus, and may not be useful, it allowed us to get some experience using our complex equation from the last section. Next we stack a number of rings on top of each other, and analyze the resultant field....
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This note was uploaded on 02/09/2012 for the course PHY PHY2053 taught by Professor Davidjudd during the Fall '10 term at Broward College.

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Fields of Rings and Coils - of length dl Fortunately dl and...

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