Calculate the magnetic field from the vector potential

In summary, the conversation discusses the derivation of radial and axial magnetic fields of a current carrying loop from its magnetic vector potential. The speaker has successfully derived the radial field but is facing difficulties with the axial field derivation. They share their derivation of the radial field and ask for help in identifying where they went wrong in the derivation of the axial field. They also inquire about the independence of certain variables in the equations and point out a potential mistake. The responder confirms the mistake and suggests a correction for equations (1) and (2).
  • #1
arjun_ar
3
4
Homework Statement
Given the magnetic vector potential of a current carrying loop in cylindrical coordinate system, derive the axial and radial magnetic fields.
Relevant Equations
Please read ahead.
I am trying to derive radial and axial magnetic fields of a current carrying loop from its magnetic vector potential. So far, I have succeeded in deriving the radial field but axial field derivation gives me trouble.
Z2N67kE.png


My derivation of radial field (eq 1) can be found here.

Can anyone point out where I went wrong in the derivation of axial field?

My derivation of axial field is given below.

EjnkUB1.png


In case the images are blurred, you can see them here and here.

On comparing, Eq.7 with Eq.2, the coefficient of E do not match.

I have done this derivation multiple times, yet arrive at the same answer.
Can anyone point me where I went wrong?
 
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  • #2
Hmm is it dead sure that K and E are independent of ##\rho,z##? So you can treat them as constants when taking the derivatives with respect to those variables?
 
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  • #3
I think the mistake is here:

1658095752159.png


The two expressions circled in green are not equivalent.

-----------------------------------------------------------------------

Do you have typographical errors in your equations (1) and (2)? Should the denominators inside the square brackets be ##(a-\rho)^2 + z^2## instead of ##(a+\rho)^2 + z^2##?
 
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  • #4
Delta2 said:
Hmm is it dead sure that K and E are independent of ##\rho,z##? So you can treat them as constants when taking the derivatives with respect to those variables?
Thank you for your response.
I think they are independent of ##\rho,z##. In my derivation of radial field, I have treated them to be independent of ##z## and arrived at the exact solution.
 
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  • #5
TSny said:
I think the mistake is here:

View attachment 304311

The two expressions circled in green are not equivalent.

-----------------------------------------------------------------------

Do you have typographical errors in your equations (1) and (2)? Should the denominators inside the square brackets be ##(a-\rho)^2 + z^2## instead of ##(a+\rho)^2 + z^2##?
Thank you! This solves my problem. I don't know how I missed it!

There is indeed a typographical error for equations (1) and (2). I will update them and provide a full derivation for axial field asap.
 
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Related to Calculate the magnetic field from the vector potential

1. How do you calculate the magnetic field from the vector potential?

The magnetic field can be calculated from the vector potential using the equation: B = ∇ x A, where B is the magnetic field and A is the vector potential.

2. What is the relationship between the magnetic field and the vector potential?

The magnetic field and the vector potential are related through the Maxwell's equations, specifically the Faraday's law of induction. The magnetic field is the curl of the vector potential, which means it is a measure of the circulation of the vector potential.

3. Can the magnetic field be calculated from the vector potential in all situations?

Yes, the magnetic field can be calculated from the vector potential in all situations, as long as the vector potential is known and the necessary mathematical operations can be performed.

4. What are the units of the magnetic field and vector potential?

The units of the magnetic field are typically measured in Tesla (T) or Gauss (G), while the units of the vector potential are typically measured in Tesla-meter (Tm) or Gauss-centimeter (Gcm).

5. Are there any practical applications of calculating the magnetic field from the vector potential?

Yes, there are many practical applications of calculating the magnetic field from the vector potential, such as in electromagnetics, magnetohydrodynamics, and quantum mechanics. It is also used in various technologies, such as MRI machines and particle accelerators.

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