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Question:
This question regards electromagnetic induction and Newton's 3rd law. Please imagine a solenoidal toroid (i.e. a donut shaped inductor) powered by an AC voltage source. It creates a changing magnetic field which is confined to the interior of the toroid (i.e. within the coils). Because of the laws of electromagnetic induction, this changing magnetic field must in turn create a changing electric field in the space surrounding the inductor. The electric field lines surrounding the inductor will be of course somewhat similar to the magnetic field lines surrounding a current carrying ring, in that they are closed upon themselves (i.e. sourceless).

Now if we place a charged object (such as a sphere) in the "donut hole" region of the inductor, this object will be accelerated back and forth through the "donut hole" by the changing electric field created by induction.

So, the electric field created by induction acts on the charged sphere (F = Eq) but the electric field of the charged sphere has nothing to act on in return. I.E. It cannot act on the source of the inductively created electric field, because there is none.

My question is this: What prevents this situation from being a violation of Newton's 3rd law?

For example, what if we clamp the charged sphere to the inductor. It seems that the whole apparatus would oscillate back and forth. More importantly, what if we connect the charged sphere to an AC voltage source which causes the magnitude of the charge on the sphere to vary in phase with the strength of the inductively created electric field? Then it seems we have a reactionless propulsion situation. Since EM induction is such a well known area of electromagnetism, my assumption is that this must not be the case.

Can you help explain the reasons why? Thankyou.



Replies:
You are not interpreting the word ``sourceless'' correctly. In the first context in which you use it, sourceless does not mean the electric field has no cause, because that would be silly. Rather it means that if you write the distant electric field as a multipole expansion there is no monopole contribution. In the language of multidimensional differential equations, a derivative field, such as the electric field, is said to have no ``sources'' when there are no poles in the corresponding integral field, such as the electric potential.

It may also be relevant that in electroSTATICS (you are dealing in electrodynamics) an electric field can have no monopole contribution only if, indeed, there are no sources (charges) within your system.

Hence you are mistaken in your conclusion that the solenoid would not oscillate via the 3rd law in synchrony with your charged sphere. Interesting question.

Grayce



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