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

Twin Wires in Space

Imagine two infinitely long, parallel wires suspended in space, separated by a distance of $2d$. Both wires carry a steady current $i$ that flows directly out of the screen towards you. Our mission is to map the magnetic field along the horizontal line connecting them.

To do this, we need our trusty tool: the Right-Hand Grip Rule. If you point your right thumb in the direction of the current (out of the screen), your curling fingers show that the magnetic field forms counter-clockwise circles around each wire. The magnitude of this field at a distance $r$ is given by Ampere's Law: $B=\frac{{\mu }_{0}i}{2\pi r}$.

A Tug of War

Let's place our origin exactly halfway between the wires. The left wire is at $x=-d$ and the right wire is at $x=+d$. Now, consider a point $x$ between them.

For the left wire, the magnetic field at this point points straight up (in the $+\hat{j}$ direction). For the right wire, the magnetic field points straight down (in the $-\hat{j}$ direction). It's a magnetic tug of war!

The net magnetic field is the vector sum:

Exactly at the center ($x=0$), the distances are equal, and the opposing fields perfectly cancel each other out. The net field is zero. But as you move closer to the left wire, its upward field grows infinitely strong, shooting the graph to $+\infty$. Conversely, as you approach the right wire, its downward field dominates, plunging the graph to $-\infty$.

United Forces

What happens outside this middle region? Let's look at a point to the far left ($x<-d$). Here, the magnetic fields from both wires point downwards. They are working together! The net field is negative, and as you get infinitely close to the left wire from the outside, it dives to $-\infty$.

Similarly, for a point to the far right ($x>d$), the magnetic fields from both wires point upwards. The net field is positive, and as you approach the right wire from the outside, it skyrockets to $+\infty$.

When we piece this all together, the graph of the magnetic field $B$ versus position $x$ reveals a beautiful symmetry. It starts negative on the far left, sweeps from positive infinity down to negative infinity in the middle, and emerges positive on the far right. This elegant curve perfectly matches the graph shown in option (a).