Question

question1:

What is the magnetic field in the axis through the center of a square current loop (width L) as a function of the height (y) above the plane of the loop?

Question 2

What is the magnetic field due to a current segment at a target point of coordinates (x,y)? Set up the integral and the cross product and integrate. No need to simplify the answer.

Question 1 What is the magnetic Seld in the axis through the center of a square current loop (width L) as a function of the height () above the plane of the loop? target point height, 2 length & wicm L current Question 2 What is the magneic field due to a current segment at a target point of coordinates (xy)? Set up the integral and the cross product and integrate. No need to simplify the answer. target port(x,y) current current, I length, L page 1

Answer 1

$Using\:Biot-Savat\:equation:$

$dB=\frac{\mu }{4\pi }\frac{I\left (d\vec{L}\times \vec{r} \right )}{r^{3}}$

$Let\:\theta \:and\:\alpha ,\:the\:angles\:respect\:the\:axis.$

$dB=\frac{\mu IdL\sin \theta }{4\pi r^{2} }=\frac{\mu IdL\sin \alpha }{4\pi r^{2}}=\frac{\mu I}{4\pi }\frac{dxR}{r^{3}}$

$Integrating:$

$B=\frac{\mu Ir}{4\pi }\int_{-\frac{L}{2}}^{\frac{L}{2}}\frac{dx}{\left ( x^{2}+R^{2} \right )^{\frac{3}{2}}}=\frac{\mu Ir}{4\pi }\frac{x}{R^{2}\left ( x^{2}+R^{2} \right )^{\frac{1}{2}}}\:|_{\frac{-L}{2}}^{\frac{L}{2}}$

$B=\frac{\mu I}{4\pi R}\left (\cos \alpha_1-\cos \alpha _2 \right )$

$and$

$R=\sqrt{y^{2}+\left (\frac{L}{2} \right )^{2}}$

$\cos \alpha _1-\cos \alpha _2=\frac{\frac{L}{2}}{\sqrt{R^{2}+\frac{L^{2}}{4}}}=\frac{L}{2\sqrt{y^{2}+\frac{L^{2}}{2}}}$

$substituting:$

$B=\frac{\mu I}{4\pi R}\frac{a}{\sqrt{y^{2}+\frac{L^{2}}{2}}}$

$This\:is\:the\:result\:for\:each\:side,\:so,\:total\:B\:is:$

$B_{t}=4B\sin \beta=4B\sin \left ( \frac{L}{2R} \right )$