Two long conductors are arranged as shown above to form overlapping cylinders, each of radius r, whose centers are separated by a distance d. Current of density J flows into the plane of the page along the shaded part of one conductor and an equal current flows out of the plane of the page along the shaded portion of the other, as shown. What are the magnitude and direction of the magnetic field at point A? :

Options

A. `(mu_(0))/(2pi) pi Jd` in the + y-direction
B. `(mu_(0))/(2pi) d^(2) (J)/(r)`, in the + y-direction
C. zero
D. None of these

Answer

A

Explanation

Let the current density in complete left cylinder is `vec(J)`, then current density in complete right cylinder is `- vec(J)`. Then magnetic field at any point P in the region of overtap is

`bar(B)=(mu_(0))/(2) bar(J)xx bar(AP) + (mu_(0))/(2)(-bar(J)xx bar(BP))`
`=(mu_(0))/(2)bar(J)xx(bar(AP)+bar(PB))+(mu_(0))/(2)(bar(J)xxbar(AB))`
Therefore magnetude of field at any point in region of overlap is `= (mu_(0))/(2)` Jd and its direction is along positive y-direction at any point P in overlap region.