Magnetic field of vector potential

In summary, the conversation revolves around the question of whether the answer obtained for a certain differentiation is equivalent to the expected answer. The conversation participant is able to successfully calculate the first two curls but struggles with the last one, eventually realizing their mistake with the help of others. The conversation ends with the participant expressing gratitude and providing further explanation for those interested.
  • #1
TheBigDig
65
2
Homework Statement
Using the vector potential A, show that the Cartesian representation of the magnetic induction field associated with a magnetic moment oriented along the Cartesian z-axis is B
Relevant Equations
[tex]\vec{B} = \nabla \times \vec{A}[/tex]
[tex]\vec{A} = \frac{\mu}{4\pi}\frac{m_z}{r^3} (-y,x,0)[/tex]
[tex]\vec{B} = \frac{\mu}{4\pi}\frac{m_z}{r^5} (3xz, 3yz, 3z^2-r^2)[/tex]
[tex]\frac{\partial}{\partial x} \frac{1}{r^3} = -\frac{3x}{r^5}[/tex]
So I was able to do out the curl in the i and j direction and got 3xz/r5 and 3yz/r5 as expected. However, when I do out the last curl, I do not get 3z2-3r2. I get the following
[tex]\frac{\partial}{\partial x} \frac{x}{(x^2+y^2+z^2)^\frac{3}{2}} = \frac{-2x^2+y^2+z^2}{(x^2+y^2+z^2)^\frac{5}{2}}[/tex]
[tex]\frac{\partial}{\partial y} \frac{-y}{(x^2+y^2+z^2)^\frac{3}{2}} = \frac{-2y^2+x^2+z^2}{(x^2+y^2+z^2)^\frac{5}{2}}[/tex]
which when added together gives me
[tex]\frac{-x^2-y^2+2z^2}{(x^2+y^2+z^2)^\frac{5}{2}}[/tex].

I can't see where I've gone wrong with this differentiation. I've tried it out on symbolab and get the same result.
 
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  • #2
Could it be that your answer is equivalent to what you are trying to get?
 
  • #3
Expand ##r^2## in terms of x y and z ...
 
  • #4
Good God, I'm a moron. Thanks to you both. I got it out there

EDIT: For those interested:
[tex]r^2=x^2+y^2+z^2[/tex]
[tex]-x^2-y^2 = z^2 -r^2[/tex]
and then sub back in
 
Last edited:

Related to Magnetic field of vector potential

1. What is a magnetic field of vector potential?

The magnetic field of vector potential is a fundamental concept in electromagnetism that describes the ability of a magnetic field to be produced by a changing electric field. It is represented by the vector potential, which is a mathematical quantity that helps us understand the behavior of the magnetic field.

2. How is the magnetic field of vector potential related to the electric field?

The magnetic field of vector potential is related to the electric field through a mathematical relationship known as the Maxwell-Faraday equation. This equation states that a changing electric field induces a magnetic field, and the strength of the induced magnetic field is proportional to the rate of change of the electric field.

3. How is the magnetic field of vector potential measured?

The magnetic field of vector potential is not directly measurable. Instead, it is calculated using the vector potential, which is derived from the electric and magnetic fields. The strength and direction of the magnetic field can then be determined using the vector potential.

4. What are the practical applications of the magnetic field of vector potential?

The magnetic field of vector potential has many practical applications, including in the design and operation of electric motors, generators, and transformers. It is also used in medical imaging techniques like magnetic resonance imaging (MRI) and in particle accelerators.

5. How does the magnetic field of vector potential affect charged particles?

Charged particles, such as electrons, are affected by the magnetic field of vector potential through the Lorentz force. This force causes the charged particles to experience a force perpendicular to their motion, which can result in circular or helical motion. This effect is crucial in many technological applications, such as particle accelerators and cathode ray tubes.

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