# Electromagnetic Waves and Inertia

I have a conceptual question. Suppose we have two bodies of equal masses. One is electrically neutral and the other is electrically charged (positivley charged). Now we apply equal magnitude of force on both of them, so with some displacement, kinetic energies of both the bodies will increase equally or unequally?

We know that accelerating charges produce electromagnetic waves, so in case 2, when the sphere is accelerated it will also produce some electromagnetic waves, the energy corresponding to those waves will come from the work done by the force itself, so the increase in kinetic energy of the mass in case 2 should be less. Is my explanation correct?

If the explanation is correct then we can say that the body in case 2 will exhibit greater inertia and the external agent trying to accelerate the body in case 2 will feel greater resistance in accelerating the body 2, is this understanding correct? If Yes, then how do we understand this "Extra Inertia" which the body 2 will exhibit. Is there a name to this "Inertia"? Is there some literature on it? Could somebody explain me some greater details in this case?

Following is the summary of the question:

1. In the shown cases, which body will acquire greater kinetic energy for the same displacement? and Why?
2. Since body 2 will also also emit some electromagnetic waves (due to accelerating charges) so can we say that body 2 will possess lesser kinetic energy in comparison to body 1, for the same displacement?
3. If point 2 is correct then can we say that body 2 will possess greater "inertia" in comparison to body 1? If yes, then can somebody explain me more details on this extra "inertia"? How do we understand it?

Thank you.

Let me first note that the two bodies in question are not identical: since charged particles are not massless, bringing extra charge will either increase the mass of the body or one would have to compensate this mass increase by removing some parts of the body.

Further, accelerated charges radiate, which means that there would be an additional force acting on a body, similar to friction or viscous force in a liquid. So the body would acquire less kinetic energy... but not because it has higher inertia, but because there are additional forces acting on it.

In some cases such phenomena are indeed described by ascribing a different mass to objects (one can even talk about renormalizing mass, although it may lead to confusion with renormalization in particle theory) - this is done for objects moving in viscous fluids, but also for particles interacting with lattice vibrations (polaron problem) or with electromagnetic field (sometimes called "polaritons", although this deviates from the standard meaning of this term).

• I would argue that because inertia/mass is object tendency to stay in motion'/ 'resistance to acceleration' , and when charges experience a force, a force always opposes it, that to me, is by definition inertia, this force opposes acceleration, which is by definition, exactly what inertia is. See 'electromagnetic mass' Commented May 10, 2022 at 8:36
• @jensenpaull this is what I described in the last paragraph - it can be viewed as a fictituous mass, but it is a force. To the extent that the concepts of force and mass are meaningful, they are not the same. Commented May 10, 2022 at 8:42

Inertia, property of a body by virtue of which it opposes any agency that attempts to put it in motion or, if it is moving, to change the magnitude or direction of its velocity. Inertia is a passive property and does not enable a body to do anything except oppose such active agents as forces and torques. A moving body keeps moving not because of its inertia but only because of the absence of a force to slow it down, change its course, or speed it up. There are two numerical measures of the inertia of a body: its mass, which governs its resistance to the action of a force, and its moment of inertia about a specified axis, which measures its resistance to the action of a torque about the same axis.

Back to basics :

We can visualize the creation of EM waves as follows. If a charged particle accelerates (moves faster, slower or changes direction), it produces both an electric field (because the particle is charged) and a magnetic field (because the particle is moving). Furthermore because the motion of the particle is changing, the electric field is changing and the magnetic field is changing.

The time sequence is as follows: acceleration, then it creates conditions for radiation. Thus if the two masses have the same value, whether charged or not, they will exhibit the same inertia.

• Thanks for your response. Does it mean that the energy which electromagnetic waves carry does not come from the work done by the force, indirectly? If Yes then what is the source of energy that EM waves carry? Commented May 10, 2022 at 6:33
• Are you sure about this answer? See 'Abraham lorentz force' and 'electromagnetic mass'. Commented May 10, 2022 at 7:15
• @DevanshMittal it does, but it happens in a second interaction level. First the force impinges on the collection of atoms, whether charged or not, and then the acceleration can emit electromagnetic waves. Commented May 10, 2022 at 7:18
• Anna, if energy is carried away from the body by EM waves, for the same force applied the charged object must move slower. For Neutral, $\int \vec{F} \cdot \vec{dr}$ causes some kinetic energy Ke, yet a charged object under that same force, will then, also have Ke, but also radiate energy. So energy for the 2 systems are different despite the same force applied. Commented May 10, 2022 at 7:24
• @jensenpaull think in terms of feynman diagrams. The force of acceleration is a different Feynman diagram vertex, the creation of photons is consequent.( Of course electromagnetic waves have invariant mass from all those different angle photons, think of the pi0). The energy is supplied by the incoming force Commented May 10, 2022 at 7:37