Animated Solution for Physics - Magnetic Effects of Current: Two long parallel wires are at a distance $2d$ apart. They carry steady equal currents flowing out of the plane of the paper as shown. The variation of the magnetic field $B$ along the line $X{X}^{\prime }$ is given by

Let's visualize the physical setup of the problem. We are given two long, parallel wires separated by a total distance of $2d$. To make our mathematical analysis elegant and symmetric, we place the origin of our coordinate system exactly halfway between the wires. This means the left wire is located at $x=-d$ and the right wire is at $x=+d$.

Both wires carry a steady current $I$, and crucially, both currents are flowing straight out of the plane of the paper towards you. Our goal is to map out how the net magnetic field $B$ varies as we travel along the x-axis from $-\infty$ to $+\infty$.

To find the magnetic field at any point on the x-axis, we rely on the fundamental formula for the magnetic field produced by a long straight wire:

Where $r$ is the perpendicular distance from the wire to the point of interest. But magnitude is only half the story; we desperately need the direction. For this, we deploy the Right-Hand Grip Rule. If you point your right thumb out of the screen (in the direction of the current), your fingers will naturally curl in a counter-clockwise direction. This tells us that the magnetic field lines form counter-clockwise concentric circles around each wire.

To construct the full graph, we must slice the x-axis into three distinct regions and analyze the superposition of the fields in each.

Region 1: To the left of both wires ($x<-d$) Imagine standing far to the left. The counter-clockwise magnetic field lines from both wires will be pointing vertically downwards (in the negative y-direction) as they pass through the x-axis. Because both fields point in the same direction, they add up to a net negative field. As you walk closer to the left wire, the distance $r$ approaches zero, causing the downward magnetic field to shoot off towards $-\infty$.

Region 2: Between the wires ($-d<x<d$) Now, step into the space between the wires. Here, a tug-of-war happens. The counter-clockwise field from the left wire points upwards (positive y-direction), while the field from the right wire points downwards (negative y-direction).

Right next to the left wire, its upward field completely dominates, so the net field starts at $+\infty$. As you move towards the center, the upward field weakens and the downward field strengthens. At the exact midpoint ($x=0$), you are equidistant from both wires. The two opposing fields perfectly cancel each other out, giving a net magnetic field of exactly zero. Continuing to the right, the downward field of the right wire takes over, plunging the net field down to $-\infty$.

Region 3: To the right of both wires ($x>d$) Finally, move to the region on the far right. Here, the counter-clockwise field from both wires points vertically upwards (in the positive y-direction). Just to the right of the second wire, the field is extremely strong and positive, starting from $+\infty$. As you walk further away towards $+\infty$, the field gradually decays back to zero.

Piecing our journey together, we are looking for a graph that exhibits the following signature: 1. Starts from $-\infty$ on the far left. 2. Jumps to $+\infty$ just inside the left wire. 3. Crosses the x-axis exactly at the origin. 4. Drops to $-\infty$ just inside the right wire. 5. Emerges from $+\infty$ on the far right and decays to zero.

Comparing this signature with the given options, it perfectly matches graph (b).