Understanding Circulation and Directionality in Vector Fields

In summary, the conversation discusses the circulation of a vector field and its dependence on the direction of integration. The use of the right hand rule is mentioned as a convention for determining the direction of circulation. However, the direction of the electric field and induced current is determined by additional information known as Lenz's law. The actual field theory for this case shows that a changing magnetic flux does not create an electric field, but rather the changing current does. The conversation also mentions Stokes' theorem and how it is related to this topic. Finally, an example of measuring the induced voltage in a loop is discussed.
  • #1
LucasGB
181
0
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)?
 

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  • #2
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.
 
  • #3
Redbelly98 said:
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?
 
  • #4
So... any thoughts?
 
  • #5
Come on, guys, this must have a simple answer, but I can't find it!
 
  • #6
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 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
 
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  • #7
LucasGB said:
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.
 
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  • #8
LucasGB said:
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.
 
  • #9
LucasGB said:
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.
 
  • #10
bjacoby said:
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
 
  • #11
clem said:
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.
 
  • #12
bjacoby said:
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
 
  • #13
clem said:
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!
 

1. What are the symptoms of a problem with circulation?

A problem with circulation can present with a variety of symptoms, including cold hands and feet, numbness or tingling in the extremities, muscle cramps, and skin discoloration. Some individuals may also experience dizziness, shortness of breath, or chest pain.

2. What causes circulation problems?

There are several factors that can contribute to circulation problems, including high blood pressure, high cholesterol, diabetes, obesity, and smoking. Other underlying health conditions, such as heart disease or peripheral artery disease, can also affect circulation.

3. How is a problem with circulation diagnosed?

A doctor may perform a physical exam and ask about a patient's medical history to diagnose a problem with circulation. Some diagnostic tests, such as blood tests, electrocardiogram (ECG), or ultrasound, may also be used to assess circulation and identify any underlying issues.

4. What are the treatment options for a problem with circulation?

The treatment for a problem with circulation depends on the underlying cause and severity of the condition. Lifestyle changes, such as quitting smoking, maintaining a healthy weight, and exercising regularly, can improve circulation. In some cases, medications or surgical procedures may be recommended.

5. Is a problem with circulation preventable?

While some factors that contribute to circulation problems, such as genetics or age, cannot be controlled, there are steps individuals can take to prevent or improve circulation. Maintaining a healthy lifestyle, managing underlying health conditions, and regularly monitoring blood pressure and cholesterol levels can help prevent circulation problems.

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