Force on a superconducting cube

[IanBerkman]

Hi everyone,

I need some help to look if I did these calculations right.Let us assume a three dimensional magnetic field:

$\vec{B}(x,y,z) = B_x(x,y,z)\hat{x} + B_y(x,y,z)\hat{y} + B_z(x,y,z)\hat{z}$

The equation for the force on a superconducting particle in a magnetic field is given by:

$\vec{F} = -\frac{2}{\mu_0}\oint[\vec{n}||\vec{B}||^2-\vec{B}(\vec{n}\cdot\vec{B})]\,\mathrm{d}S$

Where the integration takes place over the surface of the superconducting particle.

Let us assume a cube with edges of a length $2l$ and its center at $(x',y',z')$
For the front face of the cube (at $x = x'+l$ and parallel to the yz-plane) the normal vector is just $\hat{x}$

The force on this face becomes:
$F_{x1} = -\frac{2}{\mu_0}\oint[\hat{x}(B_x^2+B_y^2+B_z^2)-B_x(\hat{x}\cdot B_x\hat{x})]\,\mathrm{d}S =\\ -\frac{2}{\mu_0}\hat{x}\oint B_x^2+B_y^2+B_z^2- B_x^2\,\mathrm{d}S =\\ -\frac{2}{\mu_0}\hat{x}\oint B_y^2+B_z^2\,\mathrm{d}S$

Switching to a double integral:
$F_{x1} =-\frac{2}{\mu_0}\hat{x}\int_{y'-l}^{y'+l}\int_{z'-l}^{z'+l}B_y(x'+l,y,z)^2+B_z(x'+l,y,z)^2\,\mathrm{d}z\mathrm{d}y$

For the back face (at x = x'-l) I just replaced the normal vector with $-\hat{x}$ and end up with:
$F_{x2} = \frac{2}{\mu_0}\hat{x}\int_{y'-l}^{y'+l}\int_{z'-l}^{z'+l}B_y(x'-l,y,z)^2+B_z(x'-l,y,z)^2\,\mathrm{d}z\mathrm{d}y$

And the resulting force in the x direction is then $F_{x1}+F_{x2}$
Same story with the forces on the y and z faces.

Did I do this right? Because I want to run these equations through Mathematica but I keep ending up getting wrong plots.

Thanks in advance, Ian

 

[AndyW]

Dear Ian,

Thank you for reaching out for help with your calculations. From a quick glance, it looks like you have set up the equations correctly and your integration steps are also correct. However, it is difficult to determine if you have made any errors without seeing the specific plots or values you are getting in Mathematica.

One suggestion I have is to double check your values for the magnetic field components at each point on the surface of the cube. Make sure they are consistent and match the direction of the normal vector at that point. Additionally, check that your units are consistent throughout the calculations.

If you are still having trouble with your plots, I would recommend reaching out to a colleague or professor for assistance. It can be helpful to have a second set of eyes review your work and offer suggestions for improvement. I hope this helps and good luck with your calculations.