As an inductor gains energy, where, in space, is the additional mass being stored?
Yes, I know the mass increase of an inductor under real-world conditions is exceedingly small, but where does it take place theoretically? In the center of the winding, where the magnetic field is? In the electrons of the current? In the atoms of copper?
Who says that mass has to be stored in space? Black holes seem to suggest that infinite matter can exist in a finite point.
I don’t say that just to be flippant. I don’t see any problem with the magnetic field itself storing the energy/mass involved. Ultimately, an inductor works because it’s balancing electromotive and magnetic forces which are interchangeable in that each can produce the other. Maybe I’m making an unwarranted logical jump (electromagnetism was my weakest area in physics), but that’s not entirely unlike a photon, which consists of perpendicular electrical and magnetic waves, but no “physical” component that occupies volume. Energy/mass increases in a photon are observable as a change in wavelength rather than a change in some “physical” component.
In the magnetic field. The electrons move much too slowly for the kinetic energy of their motion to contribute an appreciable amount.
And while we’re at it, for the same question in a capacitor, the answer is the electric field.
“In space” does not mean “in volume.” It could be stored at a point. But the point’s gotta be somewhere.
How about this: how does the center of gravity change?
Or a much more thorough question: how does the gravitational field/warping of spacetime change, if graphed?
Actually, in an inductor the electromotive and magnetic forces are not balanced against each other (you might be thinking of a motor). In an ideal inductor, the magnetic field increases boundlessly as long as an electromotive force is applied. It is very much unlike an electromagnetic wave (although photons mediate static fields as well, in a way that is unclear to me).
As pointed out in an earlier link, the energy density of the magnetic field is B^2/(2*mu) where B is the magnetic field and mu is the permeability of free space. We know in special relativity that energy density and mass density are two names for the same thing, proportional to each other via E=mc^2. From general relativity, we know that spacetime curvature is caused by the stress-energy tensor, so again it is the energy density of the magnetic field that warps spacetime.
Both are very difficult to measure, of course, but the answer is as ZenBeam said: The mass gain is everywhere that the magnetic field is (which is mostly within the cylinder of the inductor, but extends everywhere).
Oh, I see. And what would happen if a 2nd gravitational field was applied? Presumably the weight of the magnetic field would be transmitted onto the electrons and copper. Is there a classical explanation for how the magnetic field can effect that force on the conductor’s electrons? (This force doesn’t seem perpendicular to their velocity, it would be all in a single direction.)
And I’m intrigued, btw, as to how photons transmit static fields. What is the frequency of the photons that do this? Do their frequencies somehow cancel eachother out to create a static field?
But the gravitational field would pull on both the inductor and the magnetic field, so there wouldn’t be any force between them. Analogous to how the Earth pulls on both the Space shuttle and the astronauts, so they feel weightless (when in orbit).
But if the Earth was held above the Sun by steel beams, the astronauts would weigh the Earth down.
On average, the frequency of the photons mediating a static field is zero.
I don’t know offhand how to calculate the force on an unmoving inductor due to the gravitational force on its magnetic field.
I’d think you’d want to transform to an inertial frame of reference, with the inductor accelerating. Then you’d need to carefully keep track of how fast the charges are moving. What had been a uniform current through the coil probably now varies around the coil if the coil axis is perpendicular to the acceleration direction. There’s probably some charge distribution in the new frame of reference, even if there wasn’t any charge in the frame of reference where the inductor was stationary.
Or you could stay in the stationary frame of reference and use general relativity instead.
The idea that the mass increases in the space occupied by the field seems wrong. My thinking is that the magnetic field is only a convenience by which we describe the electrostatic field resulting when electrons move at relativistic speeds and their ratio of population density to space is shifted by Lorenz-Einstein contraction as viewed from inside the inductor core in the reference frame of the coil. So, it’s the moving electrons that aquire new potential energy for doing Coulombic attraction or repulsion work. I’m also troubled by the idea that the field grows through space at c, and wonder if the mass that becomes newly distributed at distances further and further out is traveling at c to do it.
I must say that this is one of those topics that I think I didn’t learn well enough and didn’t use at all since I learned it, now about 30 years ago. Even so, can anybody improve the thinking I describe for me?
I think this was the prevailing view before Maxwell. As you know, electromagnetic waves have measureable energy and momentum and travel at the speed of light. Einstein taught us that energy and mass are two different names for the same thing. Maxwell’s equations teach us where the energy is. The energy per unit volume is the sum of the squares of the electric field and the magnetic field (with some coefficients in there to give it units of energy). In General Relativity, Einstein showed that the source of gravity is more than just the mass (i.e. the energy), it also comes from stress. The famous equation of GR is G = 8piT, where G describes the curvature of spacetime and T is the stress-Energy tensor.
So, we have to abandon the old notion that the fields are merely a convenience, and replace them with the notion that they are real things that can propagate on their own. They have mass and exert a gravitational force.
Curiously, in quantum mechanics we learn something even stranger, it is really the electric and magnetic potential that play the key role. For example, we can measure the effect of magnetic flux going through a loop of wire, even when the wire itself is completely shielded from the magnetic field. Quantum field theories like quantum electrodynamics and quantum chromodynamics are built with the potentials as the quantities that appear in the field equations.
Makes me wonder if there’s a simple form for the energy of the electromagnetic field in terms of A (simpler than just plugging in the equations for E and B in terms of A).
Yes there is. In QM the kinetic energy plus the magnetic energy is given by:
(1/2m)(p+eA)^2
where m is the mass, p is the momentum, e is the electron charge, and A is the magnetic vector potential. When A is zero, this reduces to p^2/(2m), otherwise known as mv^2/2, the familiar kinetic energy from high school. To add the electric potential, you just add e*phi to the total energy (Hamiltionian operator), where phi is the electric scalar potential (voltage).
I don’t know why this always gets trotted out as an example of quantum weirdness, since you can get effects from a field where the field isn’t in classical electromagnetism, too. Consider an ideal solenoid (which produces a magnetic field inside, but little or no magnetic field leakage outside) nested inside another ideal solenoid of larger diameter. If you change the current through the inner solenoid, and thus the magnetic field in it, you’ll produce a current in the wires of the outer solenoid, despite there being no magnetic field at the location of those wires.
Is that the energy of a charged particle in an electromagnetic field? How’s that work for the energy in the electric and magnetic fields themselves, where m=0?
This seems counter-intuitive, assuming either toroidal or infinitely long solenoids. Can you provide a cite?
You are absolutely right. I realized my mistake shortly after posting that message, but I was on a road trip and just returned. I gave the “Hamiltonian” for a charged particle of mass m in a magnetic field.
The energy density is proportional to the square of the curl of the magnetic vector potential.