Who foots the (magnetic) energy bill?

Question

Gravitational attraction and electrostatic attraction/repulsion are intrinsic properties of matter, any particle (electron, proton) for some unknown reason can produce KE at a distance.

But magnetic attraction/force is not an intrinsic property of matter, a charged particle generates a magnetic field/flux and a magnetic force only when it is moving: higher velocity = much higher force. The definition of KE says that it is 'work done to accelerate an object', energy spent only to make it move, and does not mention the generation of other forces/energy or doing 'extramural' work.

[Electrons moving in a current produce magnetic induction that makes an electric appliance do work, who/what is spending the necessary energy to produce such work? Voltage (the difference of potential) coming from the mains provides the energy to accelerate the electrons, not the energy to blend your fruit. Same voltage would produce equivalent v/Ke if the electron is accelerated in a vacuum (such as in a synchrotron, where there is no fruit). But this is difficult to prove.]

Edit: I will not reply to any comment. They are attributing to me statements I never made. Now, to avoid technical complications (voltage, wattage etc,) that could trick me, forget the previous example, let's consider another, simpler case:

An electron is travelling at high speed (say,0.9 C). If is moving near another electron, (proton, positron or a live wire) it can make anything move, acquire KE, it can do work. When v approaches c, the attractive magnetic force gets so great that it equals the huge electrostatic repulsion.

All the issues you have raised (intrinsic spin, magnetic field, electromagnetism etc., some comments have been deleted) are irrelevant here. If those properties exist they exist even when the electron is at rest. When the electron is at rest, it still has spin, nevertheless it is not able any more to do work.

If this is an irrefutable fact, then, how come it can do work when it is moving, where is the necessary energy coming from?

Answer 1

"if the electrons are not moving in a current, the mixer can do no work. If it is moving ,same mixer does work. (Whatever you may call the chain of command and control), who is responsible/the agent of the work done by the mixer is the electron. This is a fact. Where does the energy come from?

In the case of a motor in your kitchen appliance, the energy comes from the wall socket, which came from the nearby power plant.

Answer 2

Gravitational attraction and electrostatic attraction/repulsion are intrinsic properties of matter, any particle (electron, proton) for some unknown reason can produce KE at a distance.

This is an observational fact.

But magnetic attraction/force is not an intrinsic property of matter, a charged particle generates a magnetic field/flux and a magnetic force only when it is moving: higher velocity = much higher force.

This is one of the two ways where magnetic force has been detected :

Fundamentally, contributions to any system's magnetic moment may come from sources of two kinds: (1) motion of electric charges, such as electric currents, and (2) the intrinsic magnetism of elementary particles, such as the electron.

Thus the statement does not hold. Magnetic forces are as intrinsic as electric, just harder to model.

The definition of KE says that it is 'work done to accelerate an object', energy spent only to make it move, and does not mention the generation of other forces/energy or doing 'extramural' work.

Work-Energy Principle

$$W_\text{net} = \frac{1}{2} mv_\text{final}^2 -\frac{1}{2} mv_\text{initial}^2 $$

The change in the kinetic energy of an object is equal to the net work done on the object.

That is the statement. It does not say a lot of things that are irrelevant, to the simple conservation of energy form given above. How this is accomplished is a matter of study

Let us take your last proposition:

An electron is travelling at high speed (say,0.9 C). If is moving near another electron, (proton, positron or a live wire) it can make anything move, acquire KE, it can do work.

One cannot be changing the initial conditions randomly. Each change , for example acquire more kinetic energy or lose energy to work is a different boundary condition problem.

When v approaches c, the attractive magnetic force gets so great that it equals the huge electrostatic repulsion.

The simple problem is that the electron travels with energy v. If it is being accelerated it is a different boundary condition problem to be solved. Let us look at the simpler boundary condition problem.

Let us take two electrons traveling in parallel in vacuum at constant velocity. Their individual kinetic energy to start with is constant, no work is done . If they are close enough to feel the electrostatic potential of each other , the kinetic energy can change, and the energy is supplied by the potential energy given when the tracks were designed/forced to be close to each other.

If they are close enough and move fast enough to feel the magnetic force of the motion also

$$\vec{F}= \underset{\text{Electric force}}{q\color{red}{\vec{E}}} + \underset{\text{Magnetic force}}{q\vec{v}x\color{blue}{\vec{B}}}$$

again the energy supplied for any change in the motion/kinetic_energy of the electrons is due to the initial conditions that generated the trajectories. Those initial conditions were imposed using energy which is given up when the potential energy of the two particle system changes.

Both electric fields and magnetic fields store energy. For the electric field the energy density is

$$\eta_E = \frac{\text{energy}}{\text{volume}} = \frac{1}{2} \epsilon E^2$$

This energy density can be used to calculate the energy stored in a capacitor.

For the magnetic field the energy density is

$$\eta_B =\frac{\text{energy}}{\text{volume}} = \frac{1}{2} \frac{B^2}{\mu} $$

which is used to calculate the energy stored in an inductor.

In the simple two electron in parallel fast trajectories example, the electric field energy density is given by the charges of the two electrons and the constraint of the trajectory, and the magnetic field in addition has the initial velocity as input constraint. In interaction the potential energy of the initial two particle system is transformed to the changes in the kinetic energy of the final two particle system . The potential energy was given by the initial setup of the two particle system.

To keep electrons in a steady v velocity beam in an accelerator, energy is continuously supplied by the beam line magnets etc. One does not get the beam up to a velocity and then let it fly at steady v, because it will disperse due to the forces shown above, for this reason focusing with magnets is used at the last part of the beam as seen in the linked plot.

Energy is conserved.

Answer 3

Electrons moving in a current produce magnetic induction that makes an electric appliance do work, who/what is spending the necessary energy to produce such work? Voltage (the difference of potential) coming from the mains provides the energy to accelerate the electrons, not the energy to blend your fruit.

The kinetic energy of the electrons in the current. That is exactly the energy that blends your fruit.

Consider your example of a single electron moving at relativistic speeds- it does not, in fact, produce any magnetic attractive force- the magnetic force always acts perpendicular to the velocity of a charged particle, which means other charged particles may be influenced to move towards or away from or in random directions relative to the electron depending on their initial relative velocities. But set things up right and you can arrange for them to move in technologically convenient ways- say, in circles that drag along the bulk metallic material of an electric motor rotor with them.

However those other charged particles move, however, the force that moved them is balanced by a reaction force on that first relativistic electron. You can see where that comes from by switching to the rest frame of the relativistic electron- from its point of view, all those other charged particles are moving relativisticly and producing magnetic fields that jerk it around.

As the electron gets jerked around, it loses energy as measured in the original frame- energy which has been transferred to other charged particles mediated by the magnetic field.

So, mains voltage imparts kinetic energy to electrons in a wire, and the kinetic energy of those electrons is the energy that gets transferred by the magnetic field to other electrons in the rotor of an electric motor which drag the nuclei in the rotor along with them and blends your fruit.

Answer 4

I just want to reply to your simpler example, since it's something we can hopefully both agree on without too much confusion. You say

An electron is travelling at high speed (say,0.9 C). If is moving near another electron, (proton, positron or a live wire) it can make anything move, acquire KE, it can do work. When v approaches c, the attractive magnetic force gets so great that it equals the huge electrostatic repulsion

It's worth remembering that while a moving electron produces a magnetic field, this field will NOT affect a stationary electron. A stationary electron will not be attracted or repulsed by the moving electron (except, of course, for the repulsion caused by the electron's electric field!). So in this respect, a moving electron and a stationary electron are not so different!

Now, the moving electron can of course attract another moving electron through the magnetic field it produces. However, this attraction is perpendicular to the direction of motion, so it doesn't speed the electron up, it doesn't do work, and it doesn't change anyone's kinetic energy.

Let me know if I misunderstood anything you were trying to say with your example, that's just what I got from it.

EDITED TO ADD: Also, if we're just talking about two electrons, the attractive magnetic force will never be greater than the repulsive electric force, no matter how fast you go. As you approach $c$ the two forces will balance, but that's it! They will never actually attract each other.