Magnetic forces

Forces in thermodynamics are related to the gradient of the free energy, which represents the ability of the system to do work. The Gibbs free energy when H' and T are the independent variables isG = U − T S − μ₀H' · M. From (i),

.........(i)

the force density due to a nonuniform field acting at constant temperature on a magnetized body F_m = −∇G is

.........(1)

Using an identity for ∇(A · B) (Appendix C), the expression takes a simpler formwhen M is uniformand independent of H (∇ × M = 0), and no currents are present (∇ × H' = 0). The first term on the right is zero because the curl is zero:

..........(2)

This is known as the Kelvin force. When M is parallel to the z-direction and the change in H' is also in the z-direction, the expression is F_z = μ₀M(∂H'/∂z). In any case, the force is always in the direction of the gradient of the magnitude of the applied field. A general expression for the force density when M is not independent of H is

........(3)

Note that H in this expression is the internal field, not the applied field H', and υ = 1/d, where d is the density. Hence Mυ = σ, the specific magnetic moment per kg of sample. When this is independent of density, as it is for dilute solutions or suspensions of magnetic particles, the first term is zero and the force F_m is given by the Kelvin expression for a paramagnet with H'= H. The demagnetizing field is negligible in dilute paramagnetic solutions, but in more concentrated samples such as ferrofluids, the first term takes care of the dipole–dipole interactions.