Two long wires a distance d apart equal antiparallel currents , as shown in the figure. (a) We have to show that the magnetic field strength at P, which is equidistant from the wires, is given by

(b) We have to determine the direction of .

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In the figure two long current carrying wires have been shown. Each wire carries a current . In wire 1 the current is coming out of the page and in wire 2 the current is going into the page. Separation between the two wires is d. We will calculate the magnetic field at point P, which is at a distance R from the mid-point of the line joining the two wires. We have fixed the coordinate system by indicating the unit vectors and ;

The magnetic field at P due to the wire 1 will be to the line joining 1 to P, as shown in the figure. Using the expression for the magnetic field due to a long current carrying wire, we have

The magnetic field at P due to the wire 1 will be to the line joining 1 to P, as shown in the figure. Using the expression for the magnetic field due to a long current carrying wire, we have

Therefore, the magnetic field at P due to the current carrying wires will be

Substituting the expression for , we find

(b)

The direction of the magnetic field at P is along the vector , which is in the direction of the line to the line joining the two wires at the middle, as shown in the figure.