Maxwell Equations, Macroscopic Electromagnetism, Conservation Laws — John David Jackson
Physics — E&M · helmholtz-coils-magnetic-field
Problem
A compact circular coil of radius a, carrying a current I (perhaps N turns, each with current I/N), lies in the x-y plane with its center at the origin.
(a) By elementary means [Eq. (5.4)] find the magnetic induction at any point on the z axis.
(b) An identical coil with the same magnitude and sense of the current is located on the same axis, parallel to, and a distance b above, the first coil. With the coordinate origin relocated at the point midway between the centers of the two coils, determine the magnetic induction on the axis near the origin as an expansion in powers of z, up to z4 inclusive:
Bz=(d3μ0Ia2)[1+2d43(b2−a2)z2+16d815(b4−6b2a2+2a4)z4+⋯]
where d2=a2+b2/4.
(c) Show that, off-axis near the origin, the axial and radial components, correct to second order in the coordinates, take the form
Bz=σ0+σ2(z2−2ρ2);Bρ=−σ2zρ
(d) For the two coils in part b show that the magnetic induction on the z axis for large ∣z∣ is given by the expansion in inverse odd powers of ∣z∣ obtained from the small z expansion of part b by the formal substitution, d→∣z∣.
(e) If b=a, the two coils are known as a pair of Helmholtz coils. For this choice of geometry the second terms in the expansions of parts b and d are absent (σ2=0 in part c). The field near the origin is then very uniform. What is the maximum permitted value of ∣z∣/a if the axial field is to be uniform to one part in 104, one part in 102?