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The calculation of the magnetic field due to a current loop is developed by first finding the magnetic vector potential and then taking its divergence. The solution for all points in space requires the use of elliptic integrals; however, it is shown that for points on the axis of the loop, this solution is identical to the simpler algebraic solution for points located on the axis. The specific magnetic field of the UCLA high school plasma machine is calculated by adding the effects of the 120 current loops that generate the magnetic field of the machine. Elliptic Integrals In this section we explore two elliptic integrals and their derivatives that will be important in the calculation of the magnetic field. Magnetic Field of a Current Loop In this section we derive the equation for the magnetic field due to a circular loop of current. Magnetic Vector Potential We begin by finding the magnetic vector potential of a loop of wire with radius a by the methods of [2]. We let the center of the loop be on the z axis with the loop in a plane parallel to the xy plane and with a distance h from the x axis. Magnetic Field In this section we take the curl of (18) to obtain B. For a general vector function in cylindrical coordinates. Magnetic Field In this section we take the curl of (18) to obtain B. For a general vector function in cylindrical coordinates. Magnetic Field Inside Plasma Chamber In this section we extend the solution for one current loop (28) to the plasma chamber in the UCLA high school lab. The magnetic field of the plasma machine is made up of five groups of coils of wire that surround the chamber, see Fig. 1. Each group has a total of 24 turns (eight turns long and three turns deep). We will approximate each turn to be one current loop and add the contributions from each of the 120 loops.
Tags : elliptic integrals, magnetic vector, plasma chamber, vector function, vector potential, cylindrical coordinates, algebraic solution, circular loop, current loops, current loop, xy plane, magnetic field, divergence, coils, axis
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