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?

  • $\begingroup$ 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

1 Answer 1


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:

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

$\phi$ 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

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.