Concept understanding — Magnetic Field of a Straight Wire
Magnetic Field of a Straight Wire
A long, straight wire carrying a steady current I sets up a magnetic field that circles around it. If the current flows upward, the field lines form concentric circles in planes perpendicular to the wire — stronger close to the wire, weaker farther away. This circular pattern comes from adding up the field contributions of every moving charge in the wire, and it is the simplest current-generated field — the starting point for solenoids, toroids, and electromagnets later in the chapter.
Direction: the right-hand rule
Grip the wire with your right hand, thumb pointing along the current. Your curled fingers show the direction the field circles — clockwise when viewed along the current's direction, counter-clockwise viewed against it.
The formula
For a long straight wire, the field magnitude at a perpendicular distance r from the wire is:
B=2πrμ0I
where μ0=4π×10−7T⋅m/A is the permeability of free space. This follows from Ampere's circuital law applied to a circular Amperian loop of radius r centred on the wire, over which B is constant by symmetry:
∮B⋅dl=B(2πr)=μ0Ienclosed
Why it behaves this way
Proportional to I: more current means more moving charge, so a proportionally stronger field.
Falls off as 1/r, not 1/r2: the same total field "spreads" around a circle of circumference 2πr, so it thins out as r grows — double the distance, half the field. An infinite line source falls off more slowly than a point charge's 1/r2 electric field.
Watch out
This formula assumes an infinitely long wire (or a point close enough that the ends are effectively far away). Near the actual ends of a finite wire, the field is weaker and must be found from the Biot–Savart law directly.
Worked example
A wire carries I=5A. Find B at r=2cm=0.02m.
B=2πrμ0I=2πμ0×rI=(2×10−7)×0.025=5×10−5T
That is 50μT — comparable to Earth's own magnetic field (∼25–65 μT at the surface), which is why a nearby compass needle visibly deflects (Oersted's original 1820 observation).
Tip
Memorise μ0/2π=2×10−7T⋅m/A as one constant — it turns every straight-wire field calculation into B=(2×10−7)I/r.
The big picture
This circular, 1/r field is the building block for every other current-based field in the chapter: stack many circular loops (a solenoid) or bend the wire itself into a loop, and the same Biot–Savart origin gives the fields calculated there.
The magnetic field of a straight current-carrying wire is a must-know NCERT Class 12 Physics result, commonly searched as magnetic field due to a straight wire formula class 12 or Ampere's law straight wire derivation. This inverse-distance result is tested extensively in both CBSE boards and JEE Main/NEET physics numericals on magnetism.
Integrating the Biot-Savart law along an infinite straight wire (angle from phi1 to phi2) gives B = mu0 I sin(phi1+phi2)/(4 pi a), which becomes B=mu0 I/(2 pi a) for an infinitely long wire.
✓Final answer
B=2πaμ0I, circling the wire.
Step 1. Let YY′ be an infinite straight wire carrying current I, and P a field point at perpendicular distance a. Consider a small element dl on the wire, at angle θ to the line joining it to P.
Step 2. Using the geometry (a perpendicular dropped from one end of the element, and relating the small angle dϕ subtended at P to dl), the Biot-Savart contribution reduces to dB=4πaμ0Icosϕdϕ.
Step 3. Integrating from ϕ1 to ϕ2 (the extreme angles subtended by the wire at P): B=4πaμ0I(sinϕ1+sinϕ2).
Step 4. For an infinitely long wire, ϕ1=ϕ2=90°, giving B=4πaμ0I(1+1)=2πaμ0I, directed circling the wire by the right-hand rule.
✓Final answer
B=2πaμ0I, circling the wire.
Integrate the Biot-Savart law over the whole wire using the subtended-angle geometry, then take the infinite-wire limit.
Forgetting to take both angle limits to 90 degrees for the infinite-wire case.
Mixing up the perpendicular distance a with the distance r from the element to P.