Motional EMF for loops of wire vs conducting plates

[baseballfan_ny]

I'm in an intro E&M class, and I'm trying to distinguish between Motional EMF for loops of wire and conducting plates. This question might be kind of silly, but are Eddy currents pretty much the same thing as induced currents in a loop of wire? More specifically, what I am trying to ask is if the resistive slow down in the Motional EMF cases for a loop of wire and a conducting plate can both be explained by Lorentz forces and Joule Heating/Power loss.

(Image Credits: 8.02.3x, MITx)

In this above example of motional EMF, considering the case when the loop of wire enters the magnetic field, a counterclockwise current is induced by Lenz's Law. The right most leg of the loop will experience a Lorentz force in the -$\hat i$ direction, and this resistive force will slow the loop down. That's the explanation I've seen in class. But what I'm wondering is if this slow down could also be attributed to the Joule Heating and Power loss that occurs as an induced current flows through the wire? This would decrease the Kinetic Energy and slow it down -- is that a valid way to see it in this case? Because in my E&M classes I've never seen that explanation used for loops of wire in the Motional EMF case and it seems to always be reserved for describing Eddy currents.

(Image Credits: 8.02.3x, MITx)

In the same way, when talking about Eddy currents induced on a conducting plate, could we also describe the reason for the slowdown in terms of resistive Lorentz forces? In this above case, clockwise Eddy currents will be induced such that the downwards Eddy currents on the right are "more immersed" in the magnetic field, so they will dominate in terms of Lorentz forces. These downwards currents cause a Lorentz force in the -$\hat i$ direction, a resistive force that slows the plate down? I've never seen this explanation for Eddy currents slowing a plate down -- it is always explained in terms of Joule Heating and Kinetic Energy loss. So I'm wondering if this Lorentz force explanation is even valid?

Essentially, I'm asking if the Motional EMF example for a loop of wire can be explained in terms of Joule Heating and Power loss and if the Motional EMF example for a conducting plate can be explained in terms of Lorentz forces.

Edit: Last sentence is kind of misleading. What I mean is: Essentially, I'm asking if the resistive slow down in the Motional EMF example for a loop of wire can also be explained in terms of Joule Heating and Power loss and if the Motional EMF example for a conducting plate can also be explained in terms of Lorentz forces.

 

[kuruman]

Summary:: Can the Motional EMF example for a loop of wire can be explained in terms of Joule Heating and Power loss and the Motional EMF example for a conducting plate can be explained in terms of Lorentz forces?

Essentially, I'm asking if the Motional EMF example for a loop of wire can be explained in terms of Joule Heating and Power loss and if the Motional EMF example for a conducting plate can be explained in terms of Lorentz forces.

Quick answers: No and Yes,

You cannot explain motional emf in terms of Joules heating. The same Joules heating could be achieved by a voltage source. When you have two different causes producing the same effect, you cannot work backwards and deduce the cause by observing the effect.

You can model the conducting plane as an assembly of nested rectangular wire loops moving together in which case it should be easy to find the sum of all the Lorentz forces acting on the assembly.

 

[baseballfan_ny]

You can model the conducting plane as an assembly of nested rectangular wire loops moving together in which case it should be easy to find the sum of all the Lorentz forces acting on the assembly.

Got it.

You cannot explain motional emf in terms of Joules heating. The same Joules heating could be achieved by a voltage source. When you have two different causes producing the same effect, you cannot work backwards and deduce the cause by observing the effect.

Yeah I guess it's a pretty "forceful" (I can't really think of the right word) approach to work backwards like that. But is there a physical reason we cannot explain the slowdown of a conducting loop in terms of Joules heating? I'm not sure I completely understand what you mean in that it could be achieved by a voltage source.

 

[vanhees71]

Of course, microscopically, the heat production is caused in both cases by friction of the charge carriers (in metallic conductors like wires or plates the electrons being scattered by defects and vibrations of the crystal lattice they are moving in).

Also the cause of motion of the electrons is the same for both a voltage drop and an motional EMF along the conductor: the Lorentz force on the conduction electrons, $\vec{F}=-e (\vec{E}+\vec{v} \times \vec{B})$, where $\vec{v}$ is the velocity of the conduction electrons. For motional EMF's for usual "houshold conditions" it's save to set $\vec{v}$ just the same as the velocity of the moving wire, because the drift velocity of the conduction electrons (i.e., the velocity of the electrons as measured in the rest frame of the wire) is negligible for usual "household currents" (a few millimeters per second only!).

In my experience the difficulty in understanding such examples comes from a somewhat sloppy use of Ohm's Law. Usually it's stated in the form $\vec{j}=\sigma \vec{E}$, where $\sigma$ is the electric conductivity. This formula is an approximation though, which is well justified in most "household cases". However, the complete form of Ohm's Law is $\vec{j}=\sigma(\vec{E} + \vec{v} \times \vec{B})$. The last term takes into account also the magnetic part of the Lorentz force on the electrons due to the em. field. This leads to the Hall effect.

Now, if you have just a wire at rest with some "household current" in it, the only magnetic field present is the one due to this current itself. Since the drift velocity is so small (and that's the velocity of the electrons in this case) you can usually neglect this piece and the usual form of Ohm's Law is correct. If you however have an additional external magnetic field you have to take it into account, and this is usually treated as "the Hall effect" (though there is also the Hall effect due to the magnetic field due to the current itself, but that's small for conduction electrons in wires while it's important in plasmas).

The 2nd case you cannot neglect the "Hall term" in Ohm's Law is precisely if the conductor is moving in an external magnetic field, because here the velocity is not so small and $\vec{v} \times \vec{B}$ cannot be neglected in Ohm's Law.

 

[baseballfan_ny]

Also the cause of motion of the electrons is the same for both a voltage drop and an motional EMF along the conductor: the Lorentz force on the conduction electrons, F→=−e(E→+v→×B→), where v→ is the velocity of the conduction electrons. For motional EMF's for usual "houshold conditions" it's save to set v→ just the same as the velocity of the moving wire, because the drift velocity of the conduction electrons (i.e., the velocity of the electrons as measured in the rest frame of the wire) is negligible for usual "household currents" (a few millimeters per second only!).

In my experience the difficulty in understanding such examples comes from a somewhat sloppy use of Ohm's Law. Usually it's stated in the form j→=σE→, where σ is the electric conductivity. This formula is an approximation though, which is well justified in most "household cases". However, the complete form of Ohm's Law is j→=σ(E→+v→×B→). The last term takes into account also the magnetic part of the Lorentz force on the electrons due to the em. field. This leads to the Hall effect.

That's pretty interesting! And I definitely see how the Lorentz force plays a role in both scenarios now.

I think I might have worded my question in a misleading way, and I've went back and edited. I guess what I'm still puzzled on is if the resistive slow down in the Motional EMF example for a loop of wire can be explained in terms of Joule Heating and Power loss.

 

[vanhees71]

Sure, the Ohmic loss is by definition due to friction, i.e., Joule heating and the implied power loss. Note that a simple model to derive Ohm's law (due to Drude) uses the friction of the conduction electrons in the wire.

 

[kuruman]

I think I might have worded my question in a misleading way, and I've went back and edited. I guess what I'm still puzzled on is if the resistive slow down in the Motional EMF example for a loop of wire can be explained in terms of Joule Heating and Power loss.

The agent that causes the slow down is not the heat but the eddy currents generated according to Faraday's Law. The induced eddy currents do two jobs: (a) they produce a force $I\vec L \times \vec B$ that is opposite to the velocity; (b) they generate Joules heat in the conductor because that's what currents in conductors do.

The direction of the resistive Lorentz force is determined by the direction of the induced current density vector $\vec J$ which is determined by Lenz's law. If you reverse the direction of $\vec J$, you reverse the direction of the resistive force. Now look at the Joules heating that accompanies the generation of the eddy currents. Heat is a scalar and does not "understand" direction.

The figure below shows a rectangular loop that is partially inside a steady uniform magnetic field directed into the screen.
1. If you connect a battery to the loop so that the current runs clockwise, the loop will accelerate to the left.
2. If you connect a battery to the loop so that the current runs counterclockwise, the loop will accelerate to the left right.
3. If you disconnect the battery and push the loop to the left, there will be a resistive force to the right.
4. If you disconnect the battery and push the loop to the right, there will be a resistive force to the left.

In all 4 cases, as long as the magnitude of the current is the same, the Joules heating will be the same regardless of which way the current flows. How then can one determine which direction the resistive force ought to point merely by looking at how much Joules heating is generated?

 

[vanhees71]

For 2 the force should be to the right.

 

[kuruman]

For 2 the force should be to the right.

Good catch! I cut and pasted but forgot to edit. Thanks.

 

[baseballfan_ny]

The agent that causes the slow down is not the heat but the eddy currents generated according to Faraday's Law. The induced eddy currents do two jobs: (a) they produce a force $I\vec L \times \vec B$ that is opposite to the velocity; (b) they generate Joules heat in the conductor because that's what currents in conductors do.

The direction of the resistive Lorentz force is determined by the direction of the induced current density vector $\vec J$ which is determined by Lenz's law. If you reverse the direction of $\vec J$, you reverse the direction of the resistive force. Now look at the Joules heating that accompanies the generation of the eddy currents. Heat is a scalar and does not "understand" direction.

The figure below shows a rectangular loop that is partially inside a steady uniform magnetic field directed into the screen.
1. If you connect a battery to the loop so that the current runs clockwise, the loop will accelerate to the left.
2. If you connect a battery to the loop so that the current runs counterclockwise, the loop will accelerate to the left right.
3. If you disconnect the battery and push the loop to the left, there will be a resistive force to the right.
4. If you disconnect the battery and push the loop to the right, there will be a resistive force to the left.

View attachment 273931

In all 4 cases, as long as the magnitude of the current is the same, the Joules heating will be the same regardless of which way the current flows. How then can one determine which direction the resistive force ought to point merely by looking at how much Joules heating is generated?

That seems to have cleared it up! I get you what you meant early now.

I guess I had been confused by interpreting the slow down/magnetic braking in Eddy Currents as a result of the Kinetic Energy loss from Joules Heating (as seems to have been conveyed but not intended in my intro E&M class). But it can't really be that as heat isn't a vector. It has to be the Lorentz force on the Eddy currents. The example you gave explained that very clearly. Thank you so much!