As the title suggests, whenever there is a moving wire in a magnetic field of constant magnitude, an induced voltage is produced. Now, due to the separation of charges, which is a result of the magnetic force acting on the charges within the wire, an induced electric field is produced. The induced electric field starts on the positive charges and ends on the negative ones and is in the opposite direction of the induced current. Does this mean that it is a conservative electric field? We have been told that the induced electric field generated due to changing magnetic flux in, let us say, a loop is non-conservative since it forms closed loops, and thus has no beginning nor end. Also, it is in the same direction of the induced current that opposes the change in magnetic flux (changing magnetic field, changing surface area of the loop, etc). These characteristics or properties are the exact opposite of those of the induced electric field in a moving wire. Any help is appreciated! one more question, does non-electrostatic mean non-conservative exactly, or does it add more to the meaning?
2 Answers
Is the induced electric field generated due to the separation of charges in a moving wire in a magnetic field non-conservative?
The induced electric field generated by non-moving charges is always conservative.
$$\nabla \times \vec{E_1} = 0$$
We have been told that the induced electric field generated due to changing magnetic flux in, let us say, a loop is non-conservative since it forms closed loops
The induced electric field generated by a changing magnetic flux is never conservative. It is divergence-free.
$$\nabla \cdot \vec{E_2} = 0$$
These characteristics or properties are the exact opposite of those of the induced electric field in a moving wire.
The charges in a moving wire are (most likely) moving.
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$\begingroup$ Thanks you so much! It is all very clear now :). $\endgroup$– MOMCCommented May 13, 2021 at 23:50
The induced electric field starts on the positive charges and ends on the negative ones and is in the opposite direction of the induced current. Does this mean that it is a conservative electric field?
In the frame of the wire, if the wire moves uniformly in an uniform magnetic field, the electric field of the wire charges is conservative, because it is simply the sum of the Coulomb fields of all static charges on the wire. It is not appropriate to call this field "induced field" because this term is established as the field due to currents that also create magnetic field changing in time.
In the frame of the wire there is also the time-dependent, non-conservative electric field of the magnet. Total electric field is sum of these two electric fields.
However, in the lab frame where the wire is moving, the electric field of the wire is different: it changes in time, since the charges in motion change their distance from the point of observation. They also produce magnetic field, which also changes in time. Thus due to Faraday's law, the electric field of those charges has non-zero curl, i.e. is not conservative in the lab frame. In short, electric field of a charge distribution in the frame of the wire is conservative, but electric field of that charge in the lab frame is non-conservative.
