# Motional EMF: sign difficulties

#### incrediblyfrustrated

Hi, I'm trying to understand how a moving conductor traveling with a constant velocity in a uniform magnetic field can produce an emf of -Blv where l is the length of the conductor AND have the positive charges stack up at the top. There seems to be some sort of contradiction that the top has more positive charges but also more negative (nodal) voltage?

Here's my diagram with all my work and where I'm confused.

#### studiot

You need to consider an area to calculate the EMF, you can't just consider a line conductor moving in a magnetic field. Think of the units of B.

The conventional way to do this is to lay the conductor across a pair of parallel rail conductors, with one end shorted, and consider the rate of change of area as it travels along, creating a single turn loop of increasing area.

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#### incrediblyfrustrated

Why can't I just consider a conductor traveling through free space? Was there some flaw in my logic that you could specifically point to?

I've seen the method you have mentioned before. I have also seen the method that I am describing done by others. But just about everyone gets into a handwaving argument about what to do with the sign.

Examples of this method: https://youtu.be/Wgtw5lPKFXI?t=290 and https://www.youtube.com/watch?v=SEVWBF5ATHU&feature=youtu.be&t=420

Even if I did it using the method where I set the area equal to lw, where w increases with time, and flux is equal to Blw, so that emf = -dflux/dt = -Bldw/dt = -Blv, I still am not sure what has changed? I still have a negative emf relative to the bottom node, don't I?

#### incrediblyfrustrated

I'm not sure why you say that I have left-handed axes? Didn't I use the right hand rule to calculate the direction the positive charges go (up) when the conductor is moving to the right with the magnetic field pointing into the page?

#### studiot

OK, it was late last night for my last post and I was hurried.

Looking more closely you have got your vector products in the right orientation.

Sometimes the difficulty with electric/magnetic theory is that there are several sign convention in play so you end up with minus times minus times minus etc.

Now I have the time to put some better explanation down so starting with fig 1, there is a force F1 exerted on charged particles in the conductor. This initially moves positive particles up and/or negative particles down so that electrons (they are generally the charge carriers in conductors) move down and accumulate at the bottom, leaving a net positive charge at the top.
This is not caused by an electric field.

However as soon as some charge is separated an electric field E appears and the more the charge is separated the greater the electric field and voltage between top and bottom.

Since in your case the positive is at the top, the direction of the field and voltage is down as shown in fig 2.
You can actually measure this voltage with a voltmeter.

This electric field exerts an electrostatic force F2 on any charges in the middle of the conductor as shown in fig3
So there are now two forces acting on charges in the middle of the conductor

Very quickly indeed the two forces balance and equilibrium is reached so no further charge separation occurs.

Fig 4 shows the Force equilibrium equation such that

F1 = -F2.

The second attachment shows the effect of substituting the (negative) charge on the electron into the general charge equations using q for both F1 and F2.

These are then equated using the second fact that F1 = -F2 to arrive at the relationship

V x B = -E

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#### Pmb

PHF Hall of Fame
incrediblyfrustrated said:
Hi, I'm trying to understand how a moving conductor traveling with a constant velocity in a uniform magnetic field can produce an emf of -Blv...
I believe that this is where you ran into a problem. There is a slight mistake in the terminology you used in describing your scenario. That caused a bit of a problem in correctly stating your question.

You see, the term "emf" means electromotive force which is defined as follows: https://en.wikipedia.org/wiki/Electromotive_force
Electromotive force, also called emf (denoted
and measured in volts) is the voltage developed by any source of electrical energy such as a battery or dynamo. It is generally defined as the electrical potential for a source in a circuit. A device that supplies electrical energy is called a seat of electromotive force or emf. Emfs convert chemical, mechanical, and other forms of energy into electrical energy. The product of such a device is also know as emf.
While this is the term you used the concept you described was more along the lines of a motional EMF. You see, a motional EMF is a closed line integral around a closed circuit. In your diagram there is no closed circuit. However there will be a difference in potential between the two ends and this kind of difference in potential you can measure with a volt meter.

Note: You might wonder what kind of difference in potential you can't measure with a volt meter. Consider a spherically charged object at r = (0, 0, 0). There will be an electric field around this object and there will be a difference in potential between two points at different radii. If you placed a voltmeter between two such points it will read zero. There's a subtle difference between these two scenarios which you have to be careful about. Here's why: in the example you gave there will be no flow of current in the wire, all that exists is a difference in potential across the conductors ends and that's it. If instead you placed a conductor shaped like a square loop whose sides are parallel to the xy-axes in a magnetic field in the -z-direction where the loop is moving in the +x direction (or any direction which keeps it in the xy-plane) then there will still be no current in the conductor. However if the field exists only in the +x side of the y-axes and the loop is in motion such that the magnetic field is only in part of the loop then the flux will be changing and there will be a non-zero emf in the wire and there will now be a current.

If you worked this out using the closed integral that defines the emf in he circuit then this will make a lot more sense. I wish I could easily draw and attach a diagram.

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#### icester

There is a potential energy stored in the wire segment moving with constant velocity in magnetic field... How does one calculate the amount of excess charge based on given velocity, magnetic field strength and geometry of wire segment?

#### icester

There is a potential energy stored in the wire segment moving with constant velocity in magnetic field... How does one calculate the amount of excess charge based on given velocity, magnetic field strength and geometry of wire segment?
Text books and mathematical model seem to avoid calculation of such energy and or displaced charge amount... They simply take charge q out of the equation by means of reductions thus leaving emf independent of displaced charge... The only attempt to describe such energy I found in this document: https://www.aapt.org/docdirectory/meetingpresentations/SM15/Mungan2015.pdf