Two wires, parallel to the z axis and a distance 4r apart, carry equal currents in opposite directions, as shown in the figure. A circular cylinder of radius r and length L has its axis on the z axis midway between the wires. Using Gauss’ law for magnetism we have to calculate the net out ward magnetic flux through the half of the cylindrical surface above the x axis.

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Two wires, parallel to the z axis and a distance 4r apart, carry equal currents in opposite directions, as shown in the figure. A circular cylinder of radius r and length L has its axis on the z axis midway between the wires.

The Gauss’ law for the magnetic field states that the net outward flux of magnetic field for any closed Gaussian surface S is zero. That is

.

For calculating the net outward magnetic flux through half of the cylindrical surface above the x-axis we consider a Gaussian surface S enclosed by the xz-plane within the cylinder dividing the cylinder vertically and the half of the cylindrical surface above the x-axis (A). As the magnetic field due to the current carrying wires which are parallel to the z-axis will be circular, the contribution to the flux from the top and bottom semicircular surfaces of S will be zero.

Therefore,

Or

.

We will next calculate

.

The magnetic field due to the two long wires carrying current in opposite directions, as shown in the figure, at a distance x from the z axis will be

Therefore,

We thus find that