# Where does the magnetic field energy of a charged particle moving with a uniform velocity come from?

Consider a charged particle initially at rest with respect to an inertial frame.

Let a force act on it so that it gains a velocity 'v'.

It now produces a magnetic field that has some energy associated with it.

My question is where does this energy come from? If it comes from the work done by the force acting on the particle, does it mean that $W = ΔKE$ is not valid in this case?

• This is a hard question, because that energy is likely to diverge as you integrate closer to the particle, like the electrostatic self energy does, which then suggests that the two are tightly linked. Sep 12, 2017 at 17:36
• So basically you mean that the magnetic field energy was already existing in some form just like the electric field energy?
– User
Sep 13, 2017 at 16:44
• No. What I'm saying is that this is a hard question. Sep 13, 2017 at 16:45
• So is there a very complicated answer or no answer?
– User
Sep 13, 2017 at 16:46
• Reading your discussion maybe it's helpful to read physics.stackexchange.com/a/357141/46708 Sep 14, 2017 at 10:49

## 2 Answers

You're right that the work done on the charge is not equal to the change in the charge's kinetic energy in this case. An accelerated charged body (let's assume it's of finite size, to avoid the infinite energy problem of point charges) will have a change in its "near field", and it will send off electromagnetic radiation. The work done by this external agency on the charged body will be equal to the total energy imparted to all three of these things: $$W = \Delta KE + \Delta E_\text{near field} + E_\text{radiated}.$$

It is possible to account for this "deficit" in the resulting kinetic energy of the charge by defining a so-called radiation reaction force, which can in some sense be thought of as the force that the charged body exerts on itself as it accelerates. In this case, you still have $W = \Delta KE$, so long as you define $W$ to be the work done both by the external agency and by the radiation reaction force. However, this force has some weird properties (it's proportional to the time derivative of the acceleration, for one thing), which is why you usually don't hear about it in intro E&M classes.

• So what is the origination of the magnetic field that has some energy associated with it ?
– user170242
Nov 20, 2019 at 8:28
• @TheMonk: The easiest way to think about it is that it's created by the moving charge. The fields in turn "push back" on the charge via the radiation reaction force; and so the net work on the charge is less than the work done by the external force acting on the charge. Nov 20, 2019 at 12:33

I quote from the 1938 edition of the Admiralty's handbook of wireless and telegraphy, anonymously written but clearly from the highest level of specialist knowledge. "Magnetic field energy is clearly inertial in character, just as electrical field energy is clearly kinetic in nature, due no doubt to the the motion of electrons in time and space". The writer goes on to draw attention to analogies between electro-magnetic theories and mechanical action concepts. This
handbook refers to the aether as though it was an ongoing idea, and even refers to the "jar" as a unit of capacitance. Nevertheless, common-sense always prevails in every line, for instance making it clear that "lines" of flux are only an invented aid to grasp of concepts. I'm a great fan of this book Geoff Harding BSc. (national service REME-trained as an Anti-aircraft computer electrician, then spent my life in polymer science research. Aged 87 now Cheers! Geoff