Magnetic Field of a WireJEE Main

Concept

A current creates a magnetic field that circles around it. For a long straight wire the field lines are concentric circles whose strength falls off as $1/r$. The right-hand rule gives the direction: thumb along the current, fingers curl the way $\vec B$ points.

Key formula

Derivation

Ampère's law $\oint \vec B\cdot d\vec l = \mu_0 I_{enc}$ applied to a circular loop of radius $r$ centred on the wire exploits the symmetry: $\vec B$ is tangential and constant in magnitude on the loop.

So $B\,(2\pi r) = \mu_0 I$, giving $B = \mu_0 I/(2\pi r)$.

The $1/r$ dependence (not $1/r^2$) distinguishes a line current from a point charge. A circular loop instead gives $B = \mu_0 I/(2R)$ at its centre.

Scenarios to explore

• Magnetic Field — Field of a current-carrying wire and loops.

Real-world applications

• Electromagnets, solenoids and transformers.
• Measuring current with a magnetic compass (Oersted's experiment).
• Force between parallel wires — the definition of the ampere.

JEE exam tips

• For a loop of radius R at its centre use $B = \mu_0 I/(2R)$; for a solenoid $B = \mu_0 n I$.
• Superpose fields from multiple wires as vectors at the probe point.

Common mistakes

• Using a $1/r^2$ law (that is for point charges, not straight wires).
• Getting the field sense wrong — always apply the right-hand rule deliberately.
• Forgetting B is zero along the wire's own axis only for a symmetric loop, not a straight wire.

Exam traps to avoid

• Two antiparallel currents repel; two parallel currents attract.
• Doubling the distance halves B for a wire, but the field of a dipole falls faster.