Question

The figure shows a cross -section of along thin ribbon of width b that is carrying a uniformly disturbed total current i into the page. Calculate the magnitude and direction of the magnetic field $\overrightarrow{B}$ at a point P in the plane of the ribbon at a distance afrom its edge.

• A. $\frac{{\mu }_{0}i}{4\pi b}ln\frac{(a+b)}{a}$
• B. $\frac{{\mu }_{0}i}{2b}ln\frac{(a+b)}{a}$
• C. $\frac{{\mu }_{0}i}{2\pi b}ln\frac{(a+b)}{a}$
• D. $Zero$

Answer 1

The correct option is C

$\frac{{\mu }_{0}i}{2\pi b}ln\frac{(a+b)}{a}$

Take a small element of thickness dx at a distance x from P. The current in the element is $di=\frac{i}{b}dx$,

and magnetic at P, $dB=\frac{{\mu }_0}{2\pi }\frac{di}{x}=\frac{{\mu }_0}{2\pi }\frac{\left(\frac{i}{b}\right)dx}{x}$
The total field $B=\frac{{\mu }_0}{2\pi }\frac{i}{b}{\int }_a^{(a+b)}\frac{dx}{s}=\frac{{\mu }_0}{2\pi }\frac{i}{b}{\left|lnx\right|}_a^{(a+b)}=\frac{{\mu }_0}{2\pi }\frac{i}{b}In\frac{(a+b)}{a}$