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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.
David Hansen
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2 Answers2

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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.

It's maybe more accurate to say the electromagnetic field can store energy, and always does work in the direction of the electric field component - this is treated more formally in the covariant formulation of E&M.

anon01
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In drawing an analogy between electric fields and magnetic field, you forget that there are electric monopoles while there are none such magnetic monopoles.

That's why you cannot directly find something similar to the work done on a charge in a capacitor for n inductor.

That does not mean magnetic fields don't do work. They do work on magnetic dipoles, for instance. This is also similar to the work done on an electric dipole in a capacitor.

I hope you are now able to draw to draw the right analogy.

Gaurav
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