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Step-by-Step Solution
Step 1: Recognize the Relevant Concept
The magnetic field around a long, straight wire carrying current is given by the expression
$ B = \frac{\mu_0 I}{2 \pi r} $,
where $r$ is the perpendicular distance from the wire, $I$ is the current, and $\mu_0$ is the permeability of free space.
Step 2: Identify the Geometry
Two long parallel wires are a distance $2d$ apart. Each carries an equal steady current out of the plane of the paper. We consider the line $XX'$ that connects these two wires (i.e., passing through both wires and extending on both sides).
Step 3: Determine Directions of the Magnetic Fields
By the right-hand rule, each wire produces a magnetic field that circles around it. Along the line connecting the two wires:
Between the wires: The magnetic fields from the two wires are in opposite directions and can partially or completely cancel each other.
Outside the wires: The magnetic fields from both wires add together because in those regions they point in the same direction.
Step 4: Analyze the Net Magnetic Field Magnitude along $XX'$
Because $B$ due to each wire is inversely proportional to the distance from that wire:
Near one wire, the field from that wire is large (since $r$ is small), while the field from the other wire is weaker (since $r$ is larger). The net magnetic field is thus moderately large.
At the midpoint between the two wires (i.e., exactly halfway), the fields from each wire have equal magnitude but opposite directions, often leading to net cancellation if the currents are equal and in the same direction. Hence, $B$ can be zero or minimum there.
Far away from both wires (far to the left or right), each wire's contribution diminishes ($B \sim 1/r$). The net field becomes small at large distances.
This pattern of decreasing field magnitude with increasing distance, combined with partial or near-complete cancellation in the middle, gives the characteristic shape matching Option (1).
Step 5: Conclude the Correct Graph
From the above reasoning, the correct graph must show:
a zero (or a minimum) at the midpoint between the wires,
a rise near the wires, since $B \propto 1/r$,
and a decrease again as we move very far from either wire.
Hence, the graph that matches these features is Option (1).
