I am doing my homework and I'm really stuck on this problem. It asks for the magnetic field due to a square, lying in the $xy$-plane, of side lenght $a$ and with a electric current $I$ running throught it and I am not supposed to calculate it just in the center of the square. My professor have shown that the vector potential of a finite wire of lenght $a$, lying in the $z$-axis, is given by (in cilyndrical coordinates) $$\vec{A}=\frac{\mu_0I}{4\pi}\ln\left[\frac{z+a/2+\sqrt{\rho^2+(z+a/2)^2}}{z-a/2+\sqrt{\rho^2+(z-a/2)^2}}\right]\,\hat{z}$$ So, I was trying to use this to say that the vector potential due to the wires placed in $y=\pm a/2$ is given by $$\vec{A}=\frac{\mu_0I}{4\pi}\ln\left[\frac{x+a/2+\sqrt{z^2+(y-a/2)^2+(x+a/2)^2}}{x-a/2+\sqrt{z^2+(y-a/2)^2+(x-a/2)^2}}\right]\,\hat{x}+\frac{\mu_0I}{4\pi}\ln\left[\frac{x+a/2+\sqrt{z^2+(y+a/2)^2+(x+a/2)^2}}{x-a/2+\sqrt{z^2+(y+a/2)^2+(x-a/2)^2}}\right]\,\hat{x}$$ where I shifted $z$ with $x$ because the wires are parallel to the $x$-axis and $\rho^2=z^2+(y-a)^2$, when $y>0$ and $\rho^2=z^2+(y+a)^2$, when $y<0$ . There is, of course, a similar expression for the ones placed in $x=\pm a/2$. Then, I summed up all of the components and calculated its curl to get the magnetic field, but it does not feel right since the magnetic field at the origin is zero according to my results, and I think it should not be. I just want to ask what do you think I'm doing wrong, because I have no clue what to do. Any help will be appreciated.
1 Answer
There are several papers about magnetic field of several rectangular loops discussed in my answer here including
M. Misakian, “Equations for the magnetic field produced by one or more rectangular loops of wire in the same plane,” J. Res. Natl. Inst. Stand. Technol., vol. 105, pp. 557– 564, 2000.
Dejana Herceg, Anamarija Juhas, and Miodrag Milutinov. A Design of a Four Square Coil System for a Biomagnetic Experiment, FACTA UNIVERSITATIS (NIˇS) SER.: ELEC. ENERG. vol. 22, no.3, December 2009, 285-292.
Jiaqi Li and Shilong Jin. Magnetic Field Analysis of Rectangular Current-carrying Coil Based on ANSOFT Maxwell 3D Simulation, J. Phys.: Conf. Ser. 1168 052020,2019.
We can take from these papers expressions for magnetic field in a case of one rectangular loop in the form $\vec {B}=\frac {\mu_0J_0}{4\pi}(b_x,b_y,b_z)$, where functions $b_x,b_y, b_z$ are given by (I use picture from my code for Mathematica)
Using these expressions, we can compute magnetic field in particular case $J_0=20 A, a=b=0.0521 m$ in the plane $y=0$ and on the axis
