Magnetism and Magnetic Effects: Application Problems (1)

Question

A long straight wire carries a current of $I = 5$ A. Find the magnitude of the magnetic field at a point $r = 0.10$ m from the wire. Take $\mu_0 = 4\pi \times 10^{-7}$ T m/A.

Solution — Step by Step

Why This Works

Ampere’s law gives $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$. By symmetry, $B$ is constant on a circle of radius $r$ around the wire, so $B \cdot 2\pi r = \mu_0 I$, giving the formula above.

The factor $\mu_0/(2\pi)$ shows up in many wire-related formulas — memorise it as $2 \times 10^{-7}$ in SI units.

Alternative Method

Use the Biot-Savart law and integrate over the wire — for an infinite straight wire, you’ll recover the same $\mu_0 I / (2\pi r)$ result. Ampere’s law is faster when symmetry permits.

Common Mistake

Students sometimes use $B = \mu_0 I / (4\pi r)$ — that’s the formula for a current element via Biot-Savart, not the full wire. The full infinite wire gives $\mu_0 I / (2\pi r)$. The factor of 2 difference matters in MCQs.