Set up the geometry

Two long parallel wires are separated by distance $2d$ and both carry equal currents out of the plane.

Let the wires be placed on the $x$-axis at: $x=-d \quad \text{and} \quad x=+d$ and let the line $XX'$ be the straight line joining them and extending beyond them.

Magnetic field due to one long straight wire

For a long straight wire carrying current $I$, the magnetic field magnitude at perpendicular distance $r$ is $B=\frac{\mu_0 I}{2\pi r}$

Its direction is given by the right-hand rule.

Direction of fields on the line $XX'$

Since both currents are out of the plane, the field circles each wire anticlockwise.

Along the line joining the wires:

• To the left of both wires ($x<-d$): both fields are in the same direction, so they add.
• Between the wires ($-d<x<d$): the two fields are in opposite directions, so they subtract.
• To the right of both wires ($x>d$): both fields are again in the same direction, so they add.

Expression for net field in different regions

(i) For $x>d$

Distances from the two wires are $r_1=x-d, \qquad r_2=x+d$ So, $B=\frac{\mu_0 I}{2\pi}\left(\frac{1}{x-d}+\frac{1}{x+d}\right)$ This is positive in magnitude and tends to infinity as $x\to d^+$.

(ii) For $-d<x<d$

Here the fields oppose each other. Magnitude is $B=\frac{\mu_0 I}{2\pi}\left|\frac{1}{x+d}-\frac{1}{d-x}\right|$ More usefully, with sign, $B=\frac{\mu_0 I}{2\pi}\left(\frac{1}{x+d}-\frac{1}{d-x}\right)$ At the midpoint $x=0$, $B=\frac{\mu_0 I}{2\pi}\left(\frac{1}{d}-\frac{1}{d}\right)=0$ So the graph must cross zero exactly at the center.

Also, as we approach either wire from inside, the field magnitude becomes very large: $x\to -d^+ \Rightarrow |B|\to \infty, \qquad x\to d^- \Rightarrow |B|\to \infty$

Important features of the graph

Therefore the correct $B$ vs position graph along $XX'$ must have:

1. Vertical asymptotes at the positions of the two wires, $x=-d$ and $x=+d$.
2. A zero at the midpoint $x=0$.
3. Symmetry about the midpoint.
4. Outside the wires, the fields add; between the wires, they oppose.

Matching with the options

The option that shows:

• divergence at both wires,
• zero magnetic field at the midpoint,
• and symmetric behavior on both sides, is Option A.

Comparison with stored answer

My derived answer is A, which matches the stored correct answer.