Work Done in Changing Shape of Current Carrying Loop

In summary, the potential energy for morphing a square current-carrying loop into a circular current-carrying loop without changing its length and orientation in an external magnetic field can be calculated using the formula P=M.B, where P is the potential energy, M is the magnetic moment, and B is the magnetic field. This is found by integrating the torque MxB with respect to theta. Additionally, the work done can be computed by finding the difference in potential energies between two separate loops at a given angle theta, which can be simplified using the ratio of magnetic moments and areas for the same current.
  • #1
Anmoldeep
15
2
How would you go about calculating the work done in morphing a square current-carrying loop into a circular current-carrying loop, without change in length while maintaining the same angular orientation with an external magnetic field.

My book suggests defining P(potential energy) = M.B (dot product of magnetic moment and magnetic field)

I am familiar with the above formula for a varying angle between M and B but not for a varying magnetic moment. If it's true please help me in deriving it.
Suppose for the question
1.) Edge if Square loop is 'a'
2.) Current = I
3.) Magnetic field (Uniform and perpendicular to the plane of the loop)
 
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  • #2
The relevant factor is the ratio ##4/\pi## of the areas :smile:
 
  • #3
ergospherical said:
The relevant factor is the ratio ##4/\pi## of the areas :smile:
Thanks for that, although I wanted a deeper explanation as to why potential energy is still defined by the same way, P=M.B is found by integrating Torque =MxB w.r.t (theta), here though theta remains the same and M is changing, I have found an integral by mapping every elemental length on the square back to the circle and calculating the work done for all such elemental length's but the integral is not computable, it has truncated trigonometric terms like cos(pi/4tan(theta)-theta) that too under the square root along with other terms.
 
  • #4
Anmoldeep said:
Thanks for that, although I wanted a deeper explanation as to why potential energy is still defined by the same way, P=M.B is found by integrating Torque =MxB w.r.t (theta), here though theta remains the same and M is changing, I have found an integral by mapping every elemental length on the square back to the circle and calculating the work done for all such elemental length's but the integral is not computable, it has truncated trigonometric terms like cos(pi/4tan(theta)-theta) that too under the square root along with other terms.
I think I found the solution, the textbook directly suggested the use of PE=-M.B, however we know that
Del(PE)=M.B(1-cos(theta))=PEtheta - PE0
consider two separate loops of magnetic moment M1 and M2 (square and circle respectively)
Del(PE1)=M1.B(1-cos(theta))=PEtheta - PE0
and
Del(PE2)=M2.B(1-cos(theta))=PEtheta - PE0

choose theta = pi/2 and the PEtheta term will cancel out, subtract the remaining expressions and you get the required answer as the difference in potential energies of the loops when angle between M and B is 0
 
Last edited:
  • #5
Anmoldeep said:
Thanks for that, although I wanted a deeper explanation as to why potential energy is still defined by the same way, P=M.B is found by integrating Torque =MxB w.r.t (theta)
Recall that the moment ##\mathbf{K}## acting on the magnetic dipole is ##\mathbf{K} = \mathbf{m} \times \mathbf{B}##. If the dipole is rotated to angle ##\theta## (e.g. about an axis perpendicular to the plane containing ##\mathbf{m}## and ##\mathbf{B}##) then the work done by the magnetic field is \begin{align*}
w(\theta) = \int^{\theta} \mathbf{K} \cdot d\boldsymbol{\varphi} = -\int^{\theta} mB \sin{\varphi} d\varphi = mB \cos{\varphi} \bigg{|}^{\theta} = mB\cos{\theta} + c_0
\end{align*}##c_0## can be set arbitrarily. The potential energy is simply ##u = -w = -\mathbf{m} \cdot \mathbf{B}##.
 
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  • #6
Supplementary to the above posts, maybe combining them (?):

Ratio of mag moment = ratio of areas for same current.
Then compute ## \Delta PE ## as I think @ergospherical describes.
Then ## \Delta W = \Delta P.E. ##, W = work.
 

Related to Work Done in Changing Shape of Current Carrying Loop

What is work done in changing shape of current carrying loop?

The work done in changing the shape of a current carrying loop refers to the amount of energy required to change the shape of the loop while maintaining the same current flow. This work is done against the magnetic field created by the current in the loop.

How is the work done in changing shape of current carrying loop calculated?

The work done in changing the shape of a current carrying loop can be calculated using the formula W = -∫B·dA, where B is the magnetic field and dA is the differential area of the loop. This integral represents the sum of the work done at each point on the loop.

What factors affect the work done in changing shape of current carrying loop?

The work done in changing the shape of a current carrying loop is affected by the strength of the magnetic field, the size of the loop, and the rate at which the shape is changing. A larger loop or a stronger magnetic field will require more work to change its shape.

What is the relationship between work done and the shape of current carrying loop?

The work done in changing the shape of a current carrying loop is directly proportional to the change in shape. This means that the more the shape of the loop is changed, the more work will be required to do so.

What are the practical applications of work done in changing shape of current carrying loop?

The concept of work done in changing the shape of a current carrying loop is important in the design of electric motors and generators. It is also used in the study of electromagnetic induction and can be applied in various engineering and scientific fields.

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