In my electrodynamics class the other day we learned that inductors store energy in their magnetic fields. This raised a question in my mind: is there some mass then (or really, a density) associated with the field through E=mc^2 (neglecting the kinetic energy term.) The teacher answered that these fields do not have mass, but I didn’t really understand her reasoning (it seemed to be that E=mc^2 only applied to massive things.) Could someone explain how something can be energetic but not massive (photons seem like the obvious example)?
Someone else in the class asked a related problem, which the teacher could not answer, but I’d like to hear the answer to: Suppose we put a perfect fusion reactor in a perfectly sealed box in the middle of deep, empty space with initial momentum p[sub]i[/sub]. The fusion reactor fuses hydrogen into helium and releases energy into the box (but the energy cannot escape.) Does this imply that for momentum to be conserved (in the face of lowered mass due to the H -> He conversion) the box’s momentum must increase suddenly. In other words, to an outside observer would the box suddenly increase momentum without an apparent force acting on it?
E=mc^2 just means that mass and energy are related, not that they can’t exist in their pure forms*. The stored energy in an inductor is like the potential energy of an object raised from it’s resting place in a gravitational field. You could, theoretically, use that energy to create mass, but it’s not like the object actually “has” that potential energy as an intrinsic part of its mass-ness (if I can use that word).
Energy works the same way. It has the potential to be turned into mass, but it doesn’t need to have any mass itself.
Actually, as a general rule energy-mass equivalence is not dependant of the form of energy described. So, for example, a charged battery should have slightly greater mass than a drained battery. So I would assume the inductor described would also have greater mass.
The energy would have to be released as heat somehow. If there is an easy way for it to happen, the newly created atoms to escape (which would create thrust, so that the box AND the atoms would move in opposite directions). But you said the energy cannot escape, so I think it would most likely become thermal energy (i.e. the box would get hotter) and radiation. (If you’re doing induction now, you’ll probably get to radiation soon if you haven’t already. Basically, it’s released as light waves/radio waves). An outside observer without a thermometer might not know it, but the total energy of the box has increased and the total mass has decreased.
As for momentum, the box (as a whole) keeps its initial velocity, but loses some mass. Therefore, if I’m not mistaken, the total momentum of the box actually goes down. Perhaps then, our conclusion is that conservation of energy doesn’t necessarily apply when relativity is thrown into the works?
I’m pretty sure that’s not right. Conservation of energy is one of the most fundamental things in physics and relativity failing to protect it would be a huge problem.
The question of the box is basically, if mass is converted to energy, but does not escape, does that make a difference to an outside observer who can’t tell?
If no energy is leaked from the closed system, the outside observer would not differentiate between the energy expended from the nuclear reaction and the cumulative mass of the He nuclei. So from the outside observer’s perspective, there is no mass loss at all, and momentum is conserved.
If at some time, t, a 1 Henry inductor has a current of 1 Amp the energy stored is Li[sup]2[/sup]/2 or 1/2 Joule.
The equivalent mass according to e = mc[sup]2[/sup] is 1.67*10[sup]-17[/sup]kg.
And the energy in the field is kinetic. The equation for energy in the field is Li[sup]2[/sup]/2 which is strictly analagous to the kinetic energy of a mass, mv[sup]2[/sup]/2.
Perhaps I am missing something here, but energy and mass are equivelent. A photon has mass as most directly shown by E=mc**2. If that isn’t enough, note that a photon is effected by gravity. All energy has equivalent mass, all mass has equivalent energy. When an inductor stores energy, the mass of the inductor increases. The teacher, I guess, is technically correct the field doesn’t have mass but the energy stored in the field certainly does. In your fusion reactor problem, the box is sealed, no energy leaks out, there is no change in mass. If you heat an object you increase the mass. In your example, the increase in mass due to heating is balanced by the loss of mass due to H-> He.
Perhaps there is something in your question that I am missing, but it seems pretty obvious. (Now that Einstein has explained all of this to us ).
Obviously. Again, I failed to proof-read well. The total energy of the system does increase, but only by the same amount as mass decreases (where E=mc^2 tells us how much mass is equal to how much energy). So from a Newtonian perspective, mass and energy both changed, although really one just converted to the other.
What I MEANT to say is that perhaps conservation of MOMENTUM is violated. Is there any possibility that could be true?
There are two possible answers here, depending on what you mean by “the field has mass.” If you mean by “has mass” that it acts differently under a gravitational field — i.e., would a spring scale measure a different weight when there was energy in the fields vs. when there wasn’t — then the answer is yes. Part of special (and general) relativity is that energy and mass are different aspects of the same thing, and, in particular, both can influence and are influenced by gravity.
However, there’s another sense that physicists talk about a field “having mass”. It turns out that the fundamental fields in the Universe, like the electromagnetic field, can be viewed as the exchange of particles. The electromagnetic field is actually caused by charged particles exchanging photons with each other, and photons are of course massless. In this sense, it’s accurate to say that the electromagnetic fields of an inductor “have no mass.” (This is pretty advanced stuff, though.)
No, momentum is always conserved. What would happen would be that the reaction products (the helium nuclei, neutrons, and photons) would end up having a slight preference to be moving in the direction of motion of the reactor, instead of being sent randomly in all directions.
Another way to think about this is the principle of relativity: suppose you were moving alongside your perfect reactor in a spaceship, at exactly the same velocity. It would appear to be standing still, i.e. not increasing or decreasing its momentum.
No, not really. As the above example shows, if momentum were not conserved you’d expect things to suddenly and spontaneously accelerate, and you’d think that if particles were in the habit of doing this, we’d have noticed it by now. On a more fundamental level, conservation of momentum is related in a very fundamental way to something called “translation invariance”: the idea that the laws of physics are the same here as they are over there. So far, there has not been any serious challenge to the idea of translation invariance in physics, and most physicists would sooner gnaw their own arms off than give up translation invariance and its associated concepts.
Sorry, one small correction to the OP. In the second question I meant to ask would the VELOCITY of the box increase to compensate for the (supposedly) lowered mass (thus conserving momentum.) However it’s pretty clear to me now that it would not since the mass of the system is not actually changed.
One follow up question: Suppose I took an inductor (or a capacitor) and charged it up so to speak, so that it was now storing energy in it’s field. Then I turn on a uniform gravitational field, but anchor the inductor so that it will not move in the field. Would the magnetic field, which is not anchored but apparently has an effective mass, smear out from the effects of gravity on its mass? I suspect this question is unanswerable in the classical view that I’m learning right now, since, as MikeS noted, the field is really mediated by photons, but I’d like to check.
