Uniform magnetic field produced by three, four, and five square coils

The optimal dimensions of thin coil systems of three and four square coils for producing uniform magnetic fields are calculated. We find that for three square coils, of side d and separation s between the outer coils, the most uniform field distribution occurs with s/d=0.821 116 and with I′/I=0.512 797. I′/I is the ratio of the currents in the center coil to that of the outer coils. With four square coils, the best uniformity is obtained when a/d=0.128 106 and b/d=0.505 492, where a is the distance from the center to the inner coils and b is the distance from the center to the outer coils. The ratio of the current in the inner pair of coils to that in the outer pair must be I′/I=0.423 514. We compare the uniformity of the field produced by these coil systems with each other and with Rubens’ five-coil system, both on and off axis. It is shown that the optimal four-coil design is superior to the three- and five-coil systems. The sensitivity of the uniformity to the precision of construction is discussed. Dimensions of regions around the center of the coil systems, uniform to 1 part in 106 to 1 part in 102, are given.