Question

In figure, infinite conducting rings each having current $i$ in the direction shown are placed concentrically in the same plane as shown in the figure. The radii of rings are $r,2r,2^{2}r,2^{3}r,.....\infty$. The magnetic field at the center of the rings will be :

• A. zero
• B. $\frac{{\mu }_{0}i}{r}$
• C. $\frac{{\mu }_{0}i}{2r}$
• D. $\frac{{\mu }_{0}i}{3r}$

Answer 1

Answer: (D)

$\frac{{\mu }_{0}i}{3r}$

$B=\frac{{\mu }_{0}i}{2r}-\frac{{\mu }_{0}i}{2\left(2r\right)}+\frac{{\mu }_{0}i}{2\left(2^{2}r\right)}-......\infty$
or, $B=\frac{{\mu }_{0}i}{2r}[1-\frac{1}{2}+\frac{1}{2^2}-\frac{1}{2^3}+........\infty ]$
or, $B=\frac{{\mu }_{0}i}{2r}\left[\frac{1}{1-(-\frac{1}{2})}\right]=\frac{{\mu }_{0}i}{3r}$