Understanding Circulation and Directionality in Vector Fields

[LucasGB]

The circulation of a vector field is the closed line integral of the dot product between the vector and the infinitesimal displacement vectors along the curve. Therefore, the sign of circulation depends on which way around the curve you take the integral.

This is all very well, but this equation (attachment) establishes that the circulation of the electric field is positive when the magnetic flux is decreasing, and it is negative when the flux is increasing. But how do I know which way should I take the circulation (clockwise or anticlockwise)?

 

[Redbelly98]

We use the right hand rule, by convention: right thumb points in the direction of -∂Φ/∂t, right fingers curl/point in the direction of circulation.

 

[LucasGB]

We use the right hand rule, by convention: right thumb points in the direction of -∂Φ/∂t, right fingers curl/point in the direction of circulation.

But how can that be if -∂Φ/∂t is a scalar and, therefore, doesn't have a direction?

 

[LucasGB]

So... any thoughts?

 

[LucasGB]

Come on, guys, this must have a simple answer, but I can't find it!

 

[Bob S]

Ok try this. Let's consider a circular loop with current flowing clockwise in it. using your right thumb to point along the current, your fingers curl in the direction of the magnetic flux Φ, so the flux is pointing down through the loop. Now if you increase the current dI, the downward flux dΦ in the loop increases. If you had a secondary resistive current loop close to the primary loop, the induced counter-clockwise current (due to the minus sign=Lenz's Law) in it would increase. This secondary current is in the same direction as the electric field E in the secondary loop.

Bob S

 

[Redbelly98]

But how can that be if -∂Φ/∂t is a scalar and, therefore, doesn't have a direction?

Ah, you're right. Okay, point your right thumb in the direction of -∂B/∂t instead. Your right fingers then curl in the direction of E.

I didn't try out Bob S's suggestion, that may work too.

 

[bjacoby]

But how can that be if -∂Φ/∂t is a scalar and, therefore, doesn't have a direction?

That would be because Faraday's Law is bogus and doesn't stand on it's own. The direction of things in practical cases is determined by additional information known as Lenz's law. The various hand rules are often used to implement that law.

The actual field theory for the case shows that a changing magnetic flux does NOT create an E field! The E field is created by a changing current! So, if one has an element of current dl in a certain direction, one knows that the magnetic vector potential A will be a vector in space in the same direction as the current and falling off in intensity as 1/R where R is the distance to the current element. If the current changes that means A changes and an electric field E is then created in space related to the negative of the time rate of change of A (-dA/dt). The negative sign reflects Lenz's Law and give the correct directions and signs. In the case of a secondary wire, all the electric field directions are resolved along the direction of the wire and in the case of a source current longer than dl, one must integrate the E fields from each element of wire at the point where E is being calculated. Then E.dl is integrated along the secondary loop to get the total induced emf. OK? You can see how this works by taking apart the relation known as the Neumann formula for mutual inductance when it is used to find the emf induced in one wire due to changing current in another wire.

 

[Meir Achuz]

The circulation of a vector field is the closed line integral of the dot product between the vector and the infinitesimal displacement vectors along the curve. Therefore, the sign of circulation depends on which way around the curve you take the integral.

This is all very well, but this equation (attachment) establishes that the circulation of the electric field is positive when the magnetic flux is decreasing, and it is negative when the flux is increasing. But how do I know which way should I take the circulation (clockwise or anticlockwise)?

In Stokes' theorem, which is related to this, the direction of the normal to the open surface S is determined, by a right hand rule: If you curl the fingers of your right hand around the closed loop in the direction of integration, then you thumb gives the positive direction for the normal vector to the surface. This rule is stated in most textbooks.

 

[Bob S]

That would be because Faraday's Law is bogus and doesn't stand on it's own. The direction of things in practical cases is determined by additional information known as Lenz's law. The various hand rules are often used to implement that law.

The actual field theory for the case shows that a changing magnetic flux does NOT create an E field! The E field is created by a changing current! So, if one has an element of current dl in a certain direction, one knows that the magnetic vector potential A will be a vector in space in the same direction as the current and falling off in intensity as 1/R where R is the distance to the current element. If the current changes that means A changes and an electric field E is then created in space related to the negative of the time rate of change of A (-dA/dt)..

A while back, I built a large window-frame inductor, maybe 10" on a side. The laminated iron cross-section was about 2" by 2". On the two vertical legs, I wound 60-turns of 12-Ga wire. I put the two coils in series, and plugged it into 120V ac 60 cycles. The reactive current was a few milliamps. IWhen I wrapped an ac voltmeter leads around the top horizontal leg (the loop was orthogonal to the two excitation coils), I measured about 1 volt per turn. Wasn't the ac magnetic flux in the top horizontal leg creating the 1 volt via Faraday's Law? I also put about 1000 amps (at 1 volt) into a single turn of very heavy copper wire.

Bob S

 

[Redbelly98]

In Stokes' theorem, which is related to this, the direction of the normal to the open surface S is determined, by a right hand rule: If you curl the fingers of your right hand around the closed loop in the direction of integration, then you thumb gives the positive direction for the normal vector to the surface. This rule is stated in most textbooks.

Many responses here, but I like this one the best.

 

[Bob S]

That would be because Faraday's Law is bogus and doesn't stand on it's own. The direction of things in practical cases is determined by additional information known as Lenz's law. The various hand rules are often used to implement that law.

The actual field theory for the case shows that a changing magnetic flux does NOT create an E field! The E field is created by a changing current! So, if one has an element of current dl in a certain direction, one knows that the magnetic vector potential A will be a vector in space in the same direction as the current and falling off in intensity as 1/R where R is the distance to the current element. .

If I wind a coil, take it out in the middle of the street, and flip it around. I generate a voltage due to the roughly B = 1 Gauss field of the Earth. If the N·A (100 turns x 10 cm x 10 cm) of the coil is 1 square meter, the volt-seconds generated is ~2 x 10^-4 Tesla x 1 m² volt seconds. Where did the voltage come from? Could it possibly be Faraday's Law?

Bob S

 

[LucasGB]

In Stokes' theorem, which is related to this, the direction of the normal to the open surface S is determined, by a right hand rule: If you curl the fingers of your right hand around the closed loop in the direction of integration, then you thumb gives the positive direction for the normal vector to the surface. This rule is stated in most textbooks.

Well, that pretty much solves it. Thank you very much!