How Does the Magnetic Field Behave Along the Z-Axis of a Current Loop?

[zokomoko]

Homework Statement

A current loop (of circular shape) is located at the X-Y plane.
What is the magnetic field on the Z axis? (Use the magnetic moment and magnetic potential)

Homework Equations

M=(Ia)/c $$\widehat{z}$$

A=(MXr)/r$$^{3}$$

B=curl(A)

The Attempt at a Solution

I got that the magnetic potential's Z component is zero.
The magnetic potential's angular component is: -(Iar)/cr^3
The magnetic potential's radial component is: (Iar)/cr^3

But when I take the curl of A I get: (Ia/cr^3) $$\widehat{z}$$ , which is obviously not true.
I've gone over the calculations a few times, maybe there's something I'm missing.

Thanks in advance !

 

[anathema]

Hello!

I believe you may have made a mistake in your calculation for the magnetic potential's angular component. The correct expression should be -(Ia)/2cr^3, as shown in the equation below:

A=(Mx)/r^3 = (Ia)/cr^3

Therefore, the magnetic field on the Z axis can be found by taking the curl of A, as you did before. The correct expression for the magnetic field on the Z axis is (Ia)/cr^3 \widehat{z}.

I hope this helps! Let me know if you have any further questions.

 

[nlightner]

I would first like to clarify that the equations provided in the homework statement are incorrect. The equations for the magnetic moment and magnetic potential should be:

M = IA/c
A = (μ0/4π)∫(J(r')/|r-r'|)dV

where μ0 is the permeability of free space, J(r') is the current density at a point r', and the integral is taken over the volume of the current loop.

With that being said, let's proceed with finding the magnetic field on the Z axis using the magnetic vector potential. Since the current loop is located in the X-Y plane, we can assume that the current is flowing in the clockwise direction when viewed from the positive Z axis. Using the right-hand rule, we can see that the magnetic moment vector M points in the positive Z direction.

Next, we need to calculate the magnetic potential at a point P on the Z axis. This can be done by integrating the magnetic potential equation over the volume of the current loop. Since the current loop is located in the X-Y plane, we can use cylindrical coordinates (r, θ, z) to evaluate the integral. The magnetic potential at point P can be written as:

A = (μ0/4π)∫(Ia/cosθ)/√(r^2+z^2) dθdr

where θ ranges from 0 to 2π and r ranges from 0 to the radius of the current loop.

Evaluating this integral, we get:

A = (μ0Ia/2πc)ln(r+√(r^2+z^2))

Now, we can take the curl of A to find the magnetic field on the Z axis:

B = curl(A) = (μ0Ia/2πc)(z/(r^2+z^2))

Therefore, the magnetic field on the Z axis is given by:

B = (μ0Ia/2πc)(z/(r^2+z^2)) \widehat{z}

As a final note, it is important to always double check equations and calculations to ensure their accuracy. In this case, the incorrect equations provided in the homework statement led to incorrect results. As a scientist, it is crucial to pay attention to details and verify all calculations to ensure the validity of our findings.