Q.Magnetic field at any point at a distance R due to a long straight conductor carrying current varies as :

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:

where $\mu_0 = 4\pi \times 10^{-7}\ \text{T·m/A}$ 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:

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 $1/r^2$: the same total field "spreads" around a circle of circumference $2\pi r$, 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 $1/r^2$ electric field.

Worked example

A wire carries $I = 5\ \text{A}$. Find $B$ at $r = 2\ \text{cm} = 0.02\ \text{m}$.

$B = \frac{\mu_0 I}{2\pi r} = \frac{\mu_0}{2\pi}\times\frac{I}{r} = (2\times10^{-7})\times\frac{5}{0.02} = 5\times10^{-5}\ \text{T}$ …