Current Loop in a nonuniform magnetic field

In summary, the conversation discusses solving for the net force on a current loop in a nonuniform magnetic field. The formula F=IlxB is used and integrated around the loop to find the net force. It is important to note that the angle θ in the formula is not the same as the θ in the given diagram. Theta should remain constant as the loop has rotational symmetry.
  • #1
PEZenfuego
48
0

Homework Statement



A nonuniform magnetic field exerts a net force on a current loop of radius R. The figure shows a magnetic field that is diverging from the end of a bar magnet. The magnetic field B at the position of the current loop makes an angle θ with respect to the vertical, as the same magnitude at each point on the current loop. (I know that I need to solve in terms of R, I, B, and θ
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Homework Equations



F=IlxB

F=IlBsinθ

τ=μxB

τ=μBsinθ

The Attempt at a Solution



I fought the urge to use the force equation after substitution 2∏r for l. Instead I examined the force on one small segment and planned to integrate. The length would be in terms of arc length Δs. The I would be a constant. It seems that the problem indicates that the magnetic field B is constant (surely that is an assumption because the distance from the magnet was not indicated, right?) I also thought that the angle should be the only thing changing (I doubt this is a double integral problem). The subscript i indicates that this is for some segment i.

So

Fi=IΔsBsinθi

The first thing that jumps out at me is that we don't have the necessary Δθ, but we do have Δs. Δs=ΔθR, but this is for a different θ, right? So, here is where I got lost and thought that something was wrong.

Next I tried relating it with torque.

τ=μBsinθ=FR
τ=IABsinθi
τ=I((2∏R^2)/(θ)/2∏B))sinθi=FR

But here again we have a different θ value, correct?

I would very much appreciate some help. Thank you!
 
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  • #3
Net force
 
  • #4
Use the differential form of your F equation above, which is
d F = I d l x B for the force on an element d l of the loop.

Then integrate around the loop - an easy integration since the force is constant everywhere around the loop.
 
  • #5
I know that the answer is to be 2πRIBsinθ, but I don't see how or why.

dF=IdlxB is the same as saying dF=IΔsxB or dF=IBsinθΔs integrating yields

F=IBssinθ or F=2πRsinθ.

But doesn't the value of theta change for each segment?

Oh wait! No, it doesn't. It should remain constant as the ring has rotational symmetry. Am I correct in my reasoning here?
 
  • #6
It is important to note that in the formula F = I B Δs sinθ, θ is not the same θ as given in the diagram. Remember, in the formula, sinθ is coming from a cross product of Δs and B.
 
  • #7
PEZenfuego said:
I know that the answer is to be 2πRIBsinθ, but I don't see how or why.

dF=IdlxB is the same as saying dF=IΔsxB or dF=IBsinθΔs integrating yields

F=IBssinθ or F=2πRsinθ.

But doesn't the value of theta change for each segment?

Oh wait! No, it doesn't. It should remain constant as the ring has rotational symmetry. Am I correct in my reasoning here?

That is correct! And take note of what tsny says about theta. Keep track of angles when you take your cross-product!
 

FAQ: Current Loop in a nonuniform magnetic field

1. What is a current loop in a nonuniform magnetic field?

A current loop in a nonuniform magnetic field refers to a closed loop of wire through which an electric current flows, placed in a magnetic field that varies in strength and direction. This causes the loop to experience a force and torque, resulting in a change in its orientation.

2. What factors affect the behavior of a current loop in a nonuniform magnetic field?

The behavior of a current loop in a nonuniform magnetic field is affected by the strength and direction of the magnetic field, the current flowing through the loop, and the geometry of the loop (such as its size and shape).

3. How does a current loop in a nonuniform magnetic field produce torque?

When a current-carrying loop is placed in a nonuniform magnetic field, the magnetic field exerts a force on each segment of the loop, causing a net torque on the loop. This torque can be calculated using the cross product of the magnetic field and the current in the loop.

4. What are some real-life applications of current loops in nonuniform magnetic fields?

Current loops in nonuniform magnetic fields have various practical applications, including electric motors and generators, particle accelerators, and magnetic resonance imaging (MRI) machines. They are also used in scientific research to study the behavior of electric currents and magnetic fields.

5. How can the behavior of a current loop in a nonuniform magnetic field be controlled?

The behavior of a current loop in a nonuniform magnetic field can be controlled by changing the strength and direction of the magnetic field, altering the current flowing through the loop, or adjusting the geometry of the loop. This can be achieved using various techniques such as electromagnets, variable resistors, and changing the shape of the loop.

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