Figure shows a cross section of a long ribbon of width $ω$ that is carrying a uniformly distributed total current $i$ into the page. Calculate the magnitude and direction of the magnetic field $B$ at a point $P$ in the plane of the ribbon at a distance $d$ from its edge.
Given image

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$B = \frac{μ_{0}}{π} \frac{i}{ω} \ln (\frac{d + ω}{d})$ (upwards)

$B = \frac{μ_{0}}{2 π} \frac{i}{ω} \ln (\frac{d + ω}{d})$ (upwards)

$B = \frac{μ_{0}}{π} \frac{i}{ω} \ln (\frac{d + ω}{d})$ (downwards)

$B = \frac{μ_{0}}{2 π} \frac{i}{ω} \ln (\frac{d + ω}{d})$ (downwards)

answer is C.

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Detailed Solution

$d B = \frac{μ_{0}}{2 π} \frac{(\frac{i}{ω}) \cdot d x}{x} ∴ B = \frac{μ_{0} i}{2 πω} \int_{d}^{d + ω} \frac{d x}{x}$

$B = \frac{μ_{0}}{2 πω} \ln (\frac{d + ω}{d})$ (upwards) Ans.

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