Why Does a Voltmeter Show Zero in a Rotating Magnetic Field?

[quietrain]

Homework Statement

Consider a circular loop with resistance R. Perpendicular to the loop there is a magnetic field
B(t) = t, where t is the time. If a voltmeter is connected to two points on the loop separated
by 90 degrees, what would it show?

The Attempt at a Solution

the answer given was that it would show 0. but i don't understand how it can be 0.

if it was 0, then that would mean no potential difference across the 90degrees arc

then that means no induced current would flow right? then faraday's law is wrong?

 

[eczeno]

some pretty strange things would have to happen for it not to read zero. think of the loop as two resisters connected in parallel at points a and b, where the voltmeter is connected. now, if there is a potential difference between points a and b, say Va-Vb = V>0. well, then a current should flow from a to b, and it would split evenly through both resistors. but this is not what is happening. the current is flowing from a to b through one, and then from b to a through the other. Va-Vb would have to be V and -V at the same time! seems impossible, but, thanks to good old zero, nature has an out (she always has an out) and our voltmeter does not slip into another dimension.

 

[quietrain]

some pretty strange things would have to happen for it not to read zero. think of the loop as two resisters connected in parallel at points a and b, where the voltmeter is connected. now, if there is a potential difference between points a and b, say Va-Vb = V>0. well, then a current should flow from a to b, and it would split evenly through both resistors. but this is not what is happening. the current is flowing from a to b through one, and then from b to a through the other. Va-Vb would have to be V and -V at the same time! seems impossible, but, thanks to good old zero, nature has an out (she always has an out) and our voltmeter does not slip into another dimension.

so you mean a changing B-field will not induce any current in the loop? then wouldn't this violate faraday's law?


OR

if a current is induced in the loop, then how can the voltage between 2 points be 0? what drives the current around the loop? i thought current goes from higher voltage to lower voltage?

 

[eczeno]

no, i have seen it happen too many times to claim faraday's law could be violated. but the truth is it is not so simple as stated in elementary physics classes. there is a good reason they use the term emf instead of electric potential when discussing faraday's law. if you go to maxwell's equations (of which faraday's is one) then you will see that a changing magnetic field produces an electric field, not an electric potential.

If there were a potential difference between the two points, current would flow counterclockwise in one half of the loop, and clockwise in the other half. it doesn't do that, it flows in one of those two ways all the way around the loop. thus, there cannot be a potential difference between the points.

 

[quietrain]

no, i have seen it happen too many times to claim faraday's law could be violated. but the truth is it is not so simple as stated in elementary physics classes. there is a good reason they use the term emf instead of electric potential when discussing faraday's law. if you go to maxwell's equations (of which faraday's is one) then you will see that a changing magnetic field produces an electric field, not an electric potential.

If there were a potential difference between the two points, current would flow counterclockwise in one half of the loop, and clockwise in the other half. it doesn't do that, it flows in one of those two ways all the way around the loop. thus, there cannot be a potential difference between the points.

so there exist a circumferential electric field ,driving the electric current around the loop, because of the changing B-field right?

but this induced electric field is not because of potential differences in the loop?

so this E is not conservative? which means E =/= -∇V ?

so basically throughout the loop the potential is the same BUT there exist an electric field? this is so weird :(

 

[eczeno]

bare with me hear, i am no expert in this, so please take the following remarks with a skeptical eye.

the fact that when changing fields are present, E is no longer conservative is correct, and this means it can no longer be described by a scalar potential. the major difference is that electric field lines are closed loops in this case (i am pretty sure of this, anyway), instead of beginning and ending on charges, like the field in an electro-statics problem. imagine a vector field, E, of concentric circles centered at the origin. clearly, curl E is non-zero in this case, and so E is not conservative.

basically, when the fields start changing in time, electric potential is no longer a useful (or even meaningful, i guess) concept.

wow, i understand that a lot better now, thanks for the great question.

 

[quietrain]

bare with me hear, i am no expert in this, so please take the following remarks with a skeptical eye.

the fact that when changing fields are present, E is no longer conservative is correct, and this means it can no longer be described by a scalar potential. the major difference is that electric field lines are closed loops in this case (i am pretty sure of this, anyway), instead of beginning and ending on charges, like the field in an electro-statics problem. imagine a vector field, E, of concentric circles centered at the origin. clearly, curl E is non-zero in this case, and so E is not conservative.

basically, when the fields start changing in time, electric potential is no longer a useful (or even meaningful, i guess) concept.

wow, i understand that a lot better now, thanks for the great question.

iam starting to understand it better too :)

so the crucial point is the e-field does not start and end at charges like electrostatics!

but anyway, from faraday's law, ∇ x E = -dB/dt

a constant -dB/dt will give me a constant curl of E and hence a constant E right?

so if the induced loop E-field is constant, wouldn't the concept of potentials be viable now?

but it just seems so weird. it is telling me the potential everywhere is the same since the answer says there is no potential difference.

but then that would make the electric field lines equipotential lines . BUT equipotential lines are lines that are perpendicular to electric field lines!

 

[eczeno]

more good questions.

i think (stress on think!) that the E-field will not change in time if dB/dt is constant, but it must vary in space if its curl is non-zero.

i also think any E-field created by a changing B-field (as opposed to charges) is not representable by a scalar potential, so we cannot really talk about equipotential lines anymore, or anything to do with electric potential, it is just not applicable to this situation.

also, just thinking about the physical situation, if two points on the wire were at different potential, we would simply see different things happening than what actually happens.

what a fascinating bag of tricks nature has up her sleeve.

 

[quietrain]

more good questions.

i think (stress on think!) that the E-field will not change in time if dB/dt is constant, but it must vary in space if its curl is non-zero.

i also think any E-field created by a changing B-field (as opposed to charges) is not representable by a scalar potential, so we cannot really talk about equipotential lines anymore, or anything to do with electric potential, it is just not applicable to this situation.

also, just thinking about the physical situation, if two points on the wire were at different potential, we would simply see different things happening than what actually happens.

what a fascinating bag of tricks nature has up her sleeve.

ah... oh well thanks a lot for helping!

anyway, i found a good article , you may want to read it too(especially the last paragraph). i think you are right

http://farside.ph.utexas.edu/teaching/316/lectures/node87.html

 

[eczeno]

cheers.

i read your article (well, i skipped to the last paragraph). very interesting, now i want to know how to 'salvage' the potential!

great discussion!

 

[quietrain]

if i am not wrong, they salvage it by using vector potential A now. making use of the rule that the divergence of B = 0 , and since divergence of ( curl of A) is also 0, we can have speak of potential in terms of the vector potential A.

however, since the curl of (grad of any scalar ) is also =0,

we can choose B = curl (A +grad of any scalar)

and this will not alter the resulting B