# How can magnetic fields have energy and yet be unable to do work with that energy? [duplicate]

Magnetic fields can't do work. However, we use the following equation to describe the energy density of a magnetic field.

$u = \frac{B^2}{2\mu_0}$

The term energy density suggests that the magnetic field has the ability to do work. Is this a contradiction?

As an example, the energy stored in an inductor is

$U = \frac{1}{2}LI^2$

This energy is stored in the magnetic field of the inductor. In an RL circuit with a charged inductor, the current in the circuit continues to flow for a finite time after the battery is removed. It seems that the energy of the magnetic field in the inductor is doing work to drive the current.

I realize that as the magnetic field in the inductor drops, an electric field is created that drives the current. However, the electric field's ability to do work was derived from the energy in the magnetic field.

I also realize that the energy in the magnetic field originally came from the battery, making the battery the original source of the work done.

I am left to make the following conclusions, but I don't know if they are correct.

1. A magnetic field never directly does work.
2. A magnetic field can store the ability to do work.
3. In order for magnetic energy to be used as work, the magnetic field must transfer the energy to an entity (such as an electric field) that is able to do work directly.

## marked as duplicate by Kyle Kanos, Jim, Danu, ACuriousMind♦, user10851 Jul 9 '15 at 21:47

• That magnetic fields can't do work is a misunderstanding. They don't do work in the example of a static magnetic field and a charged particle, but one can build induction accelerators like Betatrons just fine. Almost all of your electric appliances use time varying magnetic fields to do plenty of work all the time, too. The phrase that magnetic fields don't do work "directly" is simply a misunderstanding of what a magnetic field is: an observer dependent component of an electromagnetic field that can't be separated. – CuriousOne Jul 8 '15 at 0:54

Your conclusions are pretty much correct. The simplest way to see that magnetic fields don't do work is to consider the $\mathbf F_B = q\ \mathbf v \times \mathbf B$ : that is, the force due to the magnetic field is always perpendicular to motion, so no work is done directly by magnetic field.