The Model

The magnet is shown schematically in Figure 32.1. We wish to maximize the field and its duration without destroying the windings. First, we consider destruction by magnetic loading. The field, B (units: Weber/m²), in a long solenoid is:

,

(M32.1)

where μ₀ is the permeability of air (4π × 10^-7 Wb/A.m), N is the number of turns, i is the current and is the length of the coil. The field creates a force on the current-carrying coil. It acts radially outwards, rather like the pressure in a pressure vessel, with a magnitude

.

(M32.2)

This force is a body force, not a surface force. The pressure generates a stress σ in the windings and their casing, given approximately by

,

(M32.3)

where d is the thickness of the casing. This stress must not exceed the elastic limit of the windings, and this gives the first constraint on B:

.

(M32.4)

The field is maximized by maximizing

.

(M32.5)

It is also important that the windings do not heat up too much. High-powered magnets are initially cooled in liquid nitrogen to -196°C in order to reduce the resistance of the windings. If the windings warm above room temperature, their resistance, in general, becomes too large.

The resistive energy loss in the windings during the pulse is

,

(M32.6)

where R_e is the resistance of the windings, ρ_e is the electrical resistivity of the conductor material, L is its total length, and A is its cross-section area. For a short pulse, this heat increases the temperature of the windings according to

(M32.7)

where ρ is the density of the material and C_p is its specific heat capacity.

Equating (32.6) and (32.7), and eliminating i using (32.1) gives

.

(M32.8)

Noting that the geometry of the magnet (Figure 33.1) is such that NA ≈  d, equation (M32.8) can be rearranged to give

.

(M32.9)

If the temperature is limited to 273 K, then DT ≤ 200 K. The field is maximized by maximizing

.

(M32.10)

The two conditions are independent. They are simultaneously met when the two results for B (equations M32.4 and M32.7) are equated — that is, on the coupling line

,

or

.

(M32.11)