Question

Infinite number of long straight infinite wires each carrying a current I, out of the plane of the paper located at $x={\alpha }_0,-2{\alpha }_0,3{\alpha }_0,-4{\alpha }_0,....$ upto $\infty$, on the x-axis. Another collection of infinite long straight wires each carrying current I, into the plane of the paper are located at $x={\alpha }_0,2{\alpha }_0,-3{\alpha }_0,4{\alpha }_0,....$ upto $\infty$ . ${\alpha }_0$ is a positive constant. Then magnetic field at the origin due to the above collection of wires is

• A. $0$
• B. $\frac{{\mu }_{0}l}{\pi {\alpha }_{0}ln\left(2\right)}$
• C. $\frac{{\mu }_{0}ln\left(2\right)}{\pi {\alpha }_0}$
• D. $\infty$

Answer 1

The correct option is C $\frac{{\mu }_{0}ln\left(2\right)}{\pi {\alpha }_0}$

Given structure is shown in the figure

Magnitude of magnetic field at x = 0 is
$B=\frac{2{\mu }_{0}I}{2{\alpha }_0\pi }(1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+....)=\frac{{\mu }_{0}Iln\left(2\right)}{\pi {\alpha }_0}$