Is It Incorrect to Use ∫E . dr in a Moving Conductor?

[anantchowdhary]

Using Faraday's law for a closed loop we say that the EMI is -d(phi)/dt .Now this emf is induced in the closed loop...So between which two points is the PD equal to -d(phi)/dt

Thanks

 

[tiny-tim]

Using Faraday's law for a closed loop we say that the EMI is -d(phi)/dt .Now this emf is induced in the closed loop...So between which two points is the PD equal to -d(phi)/dt

Hi anantchowdhary!

Imagine a circuit with a light bulb and nothing else.

Now cut out a bit of wire between A and B and replace it with a 12V battery.

Obviously, the PD between A and B is 12V, and it will light the bulb.

Now start again, and cut 12 bits out of the wire, and replace each of them with a 1V battery (all the same way round!).

Obviously, the PD between A and B is still 12V. But it's also 12V from one side to the other of any of the cuts.

Similarly, if you make 12,000,000 cuts each with 0.000001V

The closed loop in your question is like "infinitely" many very small batteries all the way round the loop … the PD is between any two "infinitely close" points.

I expect you got that far yourself, and then wondered how do we know which way round the voltage is?

Well, the practical answer is by using Lenz's law.

The theoretical answer is that electric PD doesn't work like that:

Electric PD isn't "conservative", like gravitational PD.

The gravitational PD between two points doesn't depend on the path taken.

The electric PD between two points does depend on the path taken.

So knowing the actual electric PD isn't enough … you must specify the path also!

 

[anantchowdhary]

umm...i appreciate your help...but i don't really understand what you are trying to say.
if the EMF induced as calculated is between any two 'infinitely close' points ,then i don't get how this answers my question.
And isn't electric PD conservative...if u see it using the integral(f.dr)...it is..

thanks

 

[cabraham]

The way I look a it is that the potential *around the closed loop* is 12V or whatever. In order to transport one coulomb of charge around the loop once along the path of the loop, you must do 12 joules of work as 1 volt = 1 joule/coulomb. Twice around the loop requires 24 coulombs. Thus the potential around the loop twice is 2 times 12 or 24 volts.

The potential associated with time varying E fields is non-conservative. The potential depends on the path taken. However, with discrete charged particles, the E field is conservative and the potential is not path dependent. In the emi case, the E field has "curl", aka "rotation" or "circulation". This makes the potential path-dependent. In the discrete charged particle case, the E field has no curl.

Have I helped or made matters worse? BR.

Claude

 

[anantchowdhary]

hehe...
i just don't understand the points betwwen which the emf is calculated...

 

[tiny-tim]

hehe...
i just don't understand the points betwwen which the emf is calculated...

It's calculated between any point and itself, once around the loop!

Remember, in E = -d(phi)/dt, phi is the flux through the entire loop, so you wouldn't expect E to depend on any particular part of the loop, would you?

Does http://members.lycos.nl/AmazingArt/E/artist7.html" help?

 

[anantchowdhary]

thanks

 

[tiny-tim]

… oops!

Does http://members.lycos.nl/AmazingArt/E/artist7.html" help?

oops! sorry, anantchowdhary!

Try http://en.wikipedia.org/wiki/Image:Escher_Waterfall.jpg instead.

 

[anantchowdhary]

um...but while solving problems in case of say a fixed square semi-loop(duno what to call it) and a rod that is free to slide...we take the EMF between the two end points of the rod...now isn't this contradictory to your statement?:O

 

[tiny-tim]

um...but while solving problems in case of say a fixed square semi-loop(duno what to call it) and a rod that is free to slide...we take the EMF between the two end points of the rod...now isn't this contradictory to your statement?:O

Hi anantchowdhary!

I'm not certain what apparatus you're referring to, but I think the answer is that the rod has a very low resistance (R), so the voltage drop (potential drop) across it (IR) is very small, so measuring the EMF between the two end points of the rod almost gives the right figure, in practice.

 

[anantchowdhary]

If we add the Potential drop across two 'infinitesmally' close points,then shoudlnt the PD across the two ends be HUGE?this is as the EMF across those close points is equal to d(phI)/dt

sorry..i know this is a bit irritating

 

[Defennder]

Hi, I think you're referring to something like qn 8 of this problem set:
http://physics.weber.edu/schroeder/phsx2220/ps7.pdf

The d(phi)/dt refers to the rate at which magnetic flux lines are being cut by the moving rod. In that case you only have to work out the rate at which the magnetic flux lines are being cut when you're given the speed of the rod. I don't see why the EMF across 2 very close points should be very huge, if anything it should be really small since the area it covers when the infinitesimal segment ds of the rod slides across the field is also small.

 

[Gear300]

If we add the Potential drop across two 'infinitesmally' close points,then shoudlnt the PD across the two ends be HUGE?this is as the EMF across those close points is equal to d(phI)/dt

sorry..i know this is a bit irritating

Not if for each of those infinitesimally close points, there is an infinitesimally small potential drop. One of Maxwell's equations for electromagnetism describes this phenomenon:
http://upload.wikimedia.org/math/7/6/9/76994d6f5c0e0285e4b229aed959500d.png
If you notice, it's a path integral.

 

[anantchowdhary]

thanks a lot everyone

 

[anantchowdhary]

Well...i have a new doubt

Please see this (the figure for the problem)
http://img80.imageshack.us/my.php?image=picjt0.jpg

Suppose that the rod ab moves with a constant speed v.Now there is a uniform magnetic field B..in the loop..

So if we want to know the PD between a and b,as tiny_tim said
we can think of a battery between two infinitesimally close points...the current in the circuit will be Blv/R(where R is the resistance of ab).

Now with two different circuits we get different answers why?i mean...we assumed the emf to be induced between any two infinitesimally close points..

 

[anantchowdhary]

Electric PD isn't "conservative", like gravitational PD.

This i assume is only for time varying fields...but isn't PD calculated at an instant...by moving a hypothetical test charge...otherwise we can't solve problems by applying ohm's law

therefore
i suppose it is calculated over time also..am i correct?

 

[cabraham]

Well...i have a new doubt

Please see this (the figure for the problem)
http://img80.imageshack.us/my.php?image=picjt0.jpg

Suppose that the rod ab moves with a constant speed v.Now there is a uniform magnetic field B..in the loop..

So if we want to know the PD between a and b,as tiny_tim said
we can think of a battery between two infinitesimally close points...the current in the circuit will be Blv/R(where R is the resistance of ab).

Now with two different circuits we get different answers why?i mean...we assumed the emf to be induced between any two infinitesimally close points..

The problem with trying to compute the current is as follows. If the system is open circuited and the voltage is measured, Voc, then the circuit is terminated in a resistance R. The current is not necessarily Voc/R. When current is present its own magnetic field opposes the original magnetic field. The potential once around the loop changes as R decreases due to the bucking action of the additional magnetic field (law of Lenz) so that "d@/dt" gets reduced, hence V is reduced. If the R value is so high so that the induced current is small enough so that its magnetic field is tiny compared with the original mag field, the V is roughly Voc, and I is roughly Voc/R. However if R is reduced eventually the induced current's mag field is large enough to reduce the net flux so as to reduce the voltage. Does this help?

Claude

 

[anantchowdhary]

im getting some of this...Could you please elaborate upon how electric field is non conservative here...and is potential difference taken as the integral including time or not...acc to me it is not over time...

thanks

 

[Gear300]

im getting some of this...Could you please elaborate upon how electric field is non conservative here...and is potential difference taken as the integral including time or not...acc to me it is not over time...

thanks

Lets say you had a circular loop of wire with a magnetic field running through the loop. The magnitude of the magnetic field is constantly changing over time and so an electric field is propagating outward due to the changing magnetic flux. This allows for a voltage to be established throughout the loop along with a current.
If the electric field here was electrostatic, then the electric field would be dependent on position rather than path. But the electric field we're focusing on here is distributed across the wire and depends on the path the wire provides. Otherwise the closed loop integral provided would be 0 in value:
http://upload.wikimedia.org/math/7/6...aed959500d.png
Also note that the electric field isn't restricted to the wire...its propagating outward.

 

[anantchowdhary]

how is possible to create such an electric field using charges?

 

[Gear300]

how is possible to create such an electric field using charges?

Have a constantly changing current...AC current can help out in this circumstance. But if you were to have a single stationary charge...then I'm thinking the field remains electrostatic (not sure of what else could happen).

 

[Defennder]

This i assume is only for time varying fields...but isn't PD calculated at an instant...by moving a hypothetical test charge...otherwise we can't solve problems by applying ohm's law

therefore
i suppose it is calculated over time also..am i correct?

Yes there are two cases, one which is time-varying and one which is steady state. Both cases are covered by one of Maxwell's equations:

$$\oint \mathbf{E} \cdot d\mathbf{r} = -\frac{\partial \varPhi_B}{\partial t}$$. When the magnetic flux is steady state, the term on the right is 0 and we have shown that the electric field is conservative, which means we can evaluate the PD by evaluating the electric potential at 2 points and taking the difference.

If it is not steady-state, then things get more complicated. We must now find the rate at which magnetic flux through some loop is changed. That value, by Faraday's law is the emf.

Please see this (the figure for the problem)
http://img80.imageshack.us/my.php?image=picjt0.jpg

Suppose that the rod ab moves with a constant speed v.Now there is a uniform magnetic field B..in the loop..
So if we want to know the PD between a and b,as tiny_tim said
we can think of a battery between two infinitesimally close points...the current in the circuit will be Blv/R(where R is the resistance of ab).
Now with two different circuits we get different answers why?i mean...we assumed the emf to be induced between any two infinitesimally close points..

I assume you mean two very close points both of which are along the length of ab and not a,b themselves. For one thing, I believe this picture may be a little misleading since considering a small segment dx of the ab rod does not take into account that it is a closed loop, which is required for emf to be generated. Think instead of the entire length ab, and the rate at which the magnetic flux through the loop is changed, given by Blv.

im getting some of this...Could you please elaborate upon how electric field is non conservative here...and is potential difference taken as the integral including time or not...acc to me it is not over time...

thanks

As said above, the electric field is non-conservative if it is induced by a time-varying magnetic flux through a loop. The potential difference is defined as the rate at which magnetic flux through the loop is changed.

 

[Defennder]

Lets say you had a circular loop of wire with a magnetic field running through the loop. The magnitude of the magnetic field is constantly changing over time and so an electric field is propagating outward due to the changing magnetic flux. This allows for a voltage to be established throughout the loop along with a current.
If the electric field here was electrostatic, then the electric field would be dependent on position rather than path. But the electric field we're focusing on here is distributed across the wire and depends on the path the wire provides. Otherwise the closed loop integral provided would be 0 in value:
http://upload.wikimedia.org/math/7/6...aed959500d.png
Also note that the electric field isn't restricted to the wire...its propagating outward.

I'm not following you here. My understanding is that the electric field would be induced in the wire which causes a current flow which in turn induces a B-field opposing the increasing B-field through the loop. What do you mean by the electric field "propagating outwards" and that it isn't restricted to the wire?

 

[Gear300]

I'm not following you here. My understanding is that the electric field would be induced in the wire which causes a current flow which in turn induces a B-field opposing the increasing B-field through the loop. What do you mean by the electric field "propagating outwards" and that it isn't restricted to the wire?

The equation doesn't really show that its restricted to the wire (I'm just considering how general the equations are). You're probably more experienced than I am with this so you might have to correct me if I'm wrong: For a constantly changing magnetic field of circular cross sectional area A1 (magnitude is changing, area is constant) and a loop of wire of circular cross sectional area A2 > A1...if it is set up so that the loop of wire has the magnetic field running through it without "touching" the wire, the voltage induced through the wire should come from an electric field propagating out from the the magnetic field so that it may influence the wire. Therefore, this propagating electric field shouldn't be restricted to the wire...shouldn't it?

 

[anantchowdhary]

Is an induced electric field present in a non stationary loop...but whose flux is changing..wrt time...?

And how can we apply Kirchoff's Loop Law in an LR circuit as there is a non conservative electric field induced in the solenoid?
and can anyone please confirm that is emf defined by
$$\oint \mathbf{E} \cdot d\mathbf{r}$$
starting from ANY point and ending at the same point on the loop
even in a moving conductor
Thanks

 

[anantchowdhary]

the field inside a conductor which is non stationary but is immersed in uniform and constant magnetic field with changing flux is non conservative is it?

 

[Defennder]

Is an induced electric field present in a non stationary loop...but whose flux is changing..wrt time...?

What is a "non-stationary" loop? If the electric/magnetic flux is time-varying then this itself would induce a time varying magnetic/electric field.

And how can we apply Kirchoff's Loop Law in an LR circuit as there is a non conservative electric field induced in the solenoid?

Kirchoff's voltage law still applies here. Although the electric field is non-conversative, the drop in potential across any circuit loop is still 0. The potential difference across the inductor is assigned that value for which the loop pd is 0. See here for a clearer explanation:
http://en.wikipedia.org/wiki/Inductance#Properties_of_inductance
http://en.wikipedia.org/wiki/Kirchoff's_voltage_law#Electric_field_and_electric_potential

and can anyone please confirm that is emf defined by
$$\oint \mathbf{E} \cdot d\mathbf{r}$$
starting from ANY point and ending at the same point on the loop
even in a moving conductor
Thanks

You don't use that equation when you're dealing with time-varying fields. Instead the more general circuit law version applies:
$$\sum^{n}_{r=1} V_r = 0$$ for any circuit loop.

 

[Defennder]

the field inside a conductor which is non stationary but is immersed in uniform and constant magnetic field with changing flux is non conservative is it?

Yes it's non-conservative. You're describing the setup of a sliding rod cutting the field lines of a constant magnetic fields, right?

 

[anantchowdhary]

yea...thats right...

 

[anantchowdhary]

You don't use that equation when you're dealing with time-varying fields. Instead the more general circuit law version applies:
$$\sum^{n}_{r=1} V_r = 0$$ for any circuit loop.

Is there anything wrong in using

$$\oint \mathbf{E} \cdot d\mathbf{r}$$

 

[Defennder]

There is nothing wrong per se, it's just that in a circuit problem, for example you're usually not given the electric field vector function so you can't use that.

 

[tiny-tim]

And how can we apply Kirchoff's Loop Law in an LR circuit as there is a non conservative electric field induced in the solenoid?
and can anyone please confirm that is emf defined by
∫E . dr
starting from ANY point and ending at the same point on the loop
even in a moving conductor

Kirchoff's voltage law still applies here. Although the electric field is non-conversative, the drop in potential across any circuit loop is still 0 …

Is there anything wrong in using
∫E . dr

Hi anantchowdhary!

Both versions are correct!

The induced emf, from Faraday's Law, is ∫E . dr, and it is measured starting from ANY point and ending at the same point on the loop.

On the other hand, the drop in potential across the loop … between the same two points … is zero.

In the first case, you're only measuring the induced emf.

In the second case, you're measuring the induced emf and the voltage drops.

If the loop is just a wire, with nothing added, then the emf is ∫E . dr, starting from any point and ending at the same point.

But the wire has resistance, so there's an IR to balance it, and that IR is also spread around the whole loop.

Now insert a light bulb, or a voltmeter, anywhere the loop … suddenly 99.999% of the resistance is now in the bulb or voltmeter, so the emf is 99.999% of E . dr, from one end of the bulb to the other.