Question

A long thin metallic strip of width ${}^{\prime }{b}^{\prime }$ carries a current $I$ along its length as shown in figure. Find the magnitude of magnetic field in the plane of the strip at a distance ${}^{\prime }{a}^{\prime }$ from its edge as shown in figure.

• A. $\frac{{\mu }_0}{4\pi }\frac{2I}{b}{\log }_e[1-\frac{b}{a}]$
• B. $\frac{{\mu }_0}{4\pi }\frac{2I}{b}{\log }_e[1+\frac{b}{a}]$
• C. $\frac{{\mu }_0}{2\pi }\frac{2I}{b}{\log }_e[1-\frac{b}{a}]$
• D. None of these

Answer 1

The correct option is B $\frac{{\mu }_0}{4\pi }\frac{2I}{b}{\log }_e[1+\frac{b}{a}]$


Current through $dx$ width $=\left(\frac{I}{b}\right)dx$

Note :- Current per unit width perpendicular to the flow is known as surface current, represented by $K$.

Magnetic field at ${}^{\prime }{\text{P}}^{\prime }$ due to this strip of width ${}^{\prime }d{x}^{\prime }$ is $B_P=\frac{{\mu }_0(\frac{I}{b}dx)}{2\pi x}$

$B=\underset{a}{\overset{a+b}{\int }}\frac{{\mu }_0(\frac{I}{b})dx}{2\pi x}$

$=\frac{{\mu }_{0}I}{2\pi b}\ln \left(\frac{a+b}{a}\right)$

Hence, option (b) is the correct answer.