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It is well known that the energy of a magnetic dipole in a magnetic field is taken as $$U = - \bf{m}.\bf{B}$$ The dipole also experiences a torque $$\bf{\tau = m \times B}$$

In classical mechanics, the torque is given as

$$\bf{\tau} = \bf{r \times F}$$ The force is derivable from a potential energy $V$, i.e. $\bf{F = -\nabla}$$V$; you can write the torque as $\bf{\tau = -r \times \nabla}$ $V$.

The dynamics of an object carrying an electric current is governed by the Lorentz force, a velocity-dependent force that cannot be derived from a potential energy function

Source: Daniel R Stump, Am J Phys 66, No. 12 December 1998 pp 1042-1043

Keeping the above quote in mind, is it permissible to use this relationship in the electromagnetic case?

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  • $\begingroup$ I'd say that it is permissible. But I'd rather write it directly as $\tau=dU/d\phi$ $\endgroup$
    – Ilja
    Apr 10, 2016 at 16:52

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A conservative force is one for which the curl is zero, i.e: force $F$ is conservative if and only if it satisfies the following condition,

$$\nabla \times F=0. $$

This is true for electrostatic forces. However, if at a point in space the magnetic field is changing with time, then the electric force will, in general, be non-conservative. The changing magnetic field induces a curl in the electric field.

The Lorentz force is given by,

$$\vec{F} = q(\vec{E} + \vec{v}×\vec{B})$$

$$\nabla \times F = q(\nabla \times E-B (\nabla. v) + (B.\nabla)v -(v.\nabla)B)$$

As you can see, in general, this will not be equal to zero. Therefore, we cannot write the Lorentz Force as the gradient of a scalar potential energy function, unless in the electrostatic case. But it can be derived from a velocity dependent potential function:

$$U= q\phi - q\vec{v}.\vec{A} $$

where
$\phi$ a scalar function o position
$v$ is the velocity of charge on which the Lorentz force law is applied and
$\vec{A}$ is the vector potential

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  • $\begingroup$ What do you mean by "can be.derived" here? - some kind of six-dimensional gradient in the position-velocity-phase-space? $\endgroup$
    – Ilja
    Apr 10, 2016 at 16:48
  • $\begingroup$ That six dimension reduces to four dimensional if you use potential formulation. The velocity dependent potentials are not conservative $\endgroup$
    – UKH
    Apr 11, 2016 at 2:21

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