Question

Equal currents $i=1$$A$ are flowing through the wires parallel to $y-$axis located at $x=+1$$m$, $x=+2$$m$, $x=+4$$m$ and so on...., etc. but in opposite directions as shown in the figure. The magnetic field (in Tesla) at origin would be:

• A. $-1.33\times {10}^{-7}$ $\hat{k}$
• B. $1.33\times {10}^{-7}$ $\hat{k}$
• C. $2.67\times {10}^{-7}$ $\hat{k}$
• D. $-2.67\times {10}^{-7}$ $\hat{k}$

Answer 1

Answer: (B)

$1.33\times {10}^{-7}$ $\hat{k}$

$\overrightarrow{B}=\frac{{\mu }_0}{2\pi }[(\frac{1}{1}+\frac{1}{4}+\frac{1}{16}+......)\hat{k}-(\frac{1}{2}+\frac{1}{8}+....)\hat{k}]$
$=\frac{{\mu }_0}{2\pi }[\left(\frac{1}{1-\frac{1}{4}}\right)\hat{k}-\frac{1}{2}\left(\frac{1}{1-\frac{1}{4}}\right)\hat{k}]$
$=\frac{2}{3}\frac{{\mu }_0}{2\pi }\hat{k}=(\frac{2}{3}\times 2\times {10}^{-7})\hat{k}=1.33\times {10}^{-7}\hat{k}$