Loop of current rising in magnetic field

In summary: Your Name]In summary, the conversation discusses a scenario where it is tempting to believe that the magnetic field is doing work on a current-carrying loop, but in reality, it is the battery or source of emf that is doing the work. The confusion arises from the use of velocities inside the integral, which are not constant and cannot be taken outside. Instead, the integral is used to calculate the total work done by the battery in the scenario. As scientists, it is important to critically evaluate and question information presented to us.
  • #1
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I'm reading through my text (Griffiths Electrodynamics), and I'm working through an example in which he tries to dismantle a scenario where it is tempting to think that the magnetic field does work.

The scenario is as follows, we have a square current carrying loop of wire of constant current I, of side width a, in a magnetic field pointing into the page B. The current circulates clockwise so that the magnitude of the force points up.

He says that if the current is set so that the magnitude of the magnetic force exceeds the gravitational force, the loop will move up(it is important to note that only the top side of the loop of wire is inside the B field here, and since both the left and right side are inside it, their force vectors cancel each other), and it is tempting to think that it is the magnetic force doing the work here, where the equation that would describe it would be:

W = IaBh

where h is its vertical displacement.

Instead he argues that it must be a battery (or any source of emf) doing the work. His argument is as follows:

When the loop begins to move up, the charges moving through it it acquire a vertical velocity u, in addition to their horizontal velocity w (pointing right). This makes their net velocity vector point diagonally up and to the right. Since the magnetic force vector is always perpendicular to the velocity vector, the force vector points diagonally up and to the left, giving it a component which points directly opposite to the horizontal velocity component.

The horizontal component of magnetic force can be written as:

FH = λauB (pointing left)

The vertical component of magnetic force can be written as:

FV = λawB = IBa (pointing up)

where λ is the linear charge density of the wire.

He then completes this argument by saying

"In a time dt, the charges move a (horizontal) distance wdt, so the work done by this agency (battery) is:

WB = λaB∫uwdt = IBah

Proving that the equation we arrived at erroneously is actually due to the emf, not the magnetic field.

My confusion is this:

Why are the two velocities inside the integral? This suggests that they are not constant, and then I'm not able to figure out how these two quantities actually are equal to each other.

If the velocities are constants, then both can be taken outside the integral, and the substiution λawB = IBa can be made, and u∫dt would simply equal h.

Thoughts?
 
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  • #2


Hello,

Thank you for bringing this scenario to our attention. it is important for us to critically evaluate and question the information presented to us, especially in textbooks. After reading through the forum post and your confusion, I would like to offer my thoughts on the matter.

Firstly, I agree with your understanding that the work done by the magnetic field in this scenario is zero. This is because the magnetic force is always perpendicular to the velocity of the charges, and therefore, does not do any work on the charges. As you mentioned, it is the battery or source of emf that is doing the work on the charges, allowing them to move against the gravitational force.

Now, to address your confusion regarding the velocities inside the integral, it is important to note that the velocities are not constant in this scenario. As the loop begins to move up, the charges acquire a vertical velocity u in addition to their horizontal velocity w. This means that the velocities are changing with time, and therefore, cannot be taken outside the integral. The integral is used to calculate the total work done by the battery, taking into account the changing velocities of the charges.

I hope this helps clarify your confusion. As scientists, it is important for us to critically analyze and question information presented to us, and I encourage you to continue doing so. Keep exploring and seeking answers to your questions.
 

1. What is the concept behind a loop of current rising in a magnetic field?

The concept behind a loop of current rising in a magnetic field is known as electromagnetic induction. This occurs when a changing magnetic field induces an electric current in a conductor, such as a loop of wire.

2. How does the direction of the current in the loop affect its interaction with the magnetic field?

The direction of the current in the loop determines the direction of the force exerted on the loop by the magnetic field. This is described by the right-hand rule, where the thumb points in the direction of the current and the fingers curl in the direction of the magnetic field.

3. What factors influence the strength of the loop of current rising in a magnetic field?

The strength of the current in the loop, the strength of the magnetic field, and the angle between the current and the magnetic field all influence the strength of the interaction between the loop and the field.

4. Can a loop of current rising in a magnetic field be used to generate electricity?

Yes, a loop of current rising in a magnetic field is the basis for many electrical generators. The changing magnetic field induces a current in the loop, which can then be used to power electrical devices.

5. Are there any real-world applications of a loop of current rising in a magnetic field?

Yes, electromagnetic induction is used in a variety of real-world applications, including generators, motors, transformers, and induction cooktops. It is also used in devices such as metal detectors and MRI machines.

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