Magnetic Field Around a Current-Carrying Wire

A long straight wire carrying current $I$ creates a magnetic field that forms concentric circles around the wire. The field's direction is given by the right-hand rule: point your right thumb along the current, and your fingers curl in the direction of the field.

The magnitude at distance $r$ from the wire is:

$B = \dfrac{\mu_0\,I}{2\pi r}$

where $\mu_0 = 4\pi \times 10^{-7}\;\text{T}\cdot\text{m/A}$ is the permeability of free space. Notice this is a $1/r$ falloff — slower than the $1/r^{2}$ of an electric field, because the wire extends infinitely along its length.

Step-by-Step Solution

Given: A wire carrying $I = 10\;\text{A}$.

Find: The magnetic field magnitude at distances $r = 0.01\;\text{m}$, $0.1\;\text{m}$, and $1\;\text{m}$.

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Step 1 — At $r = 0.01\;\text{m}$ (1 cm).

$B = \dfrac{\mu_0\,I}{2\pi r} = \dfrac{(4\pi \times 10^{-7})(10)}{2\pi(0.01)}$

The $\pi$'s cancel:

$= \dfrac{(4)(10)(10^{-7})}{2(0.01)}$

$= \dfrac{40 \times 10^{-7}}{0.02}$

$= 2 \times 10^{-4}\;\text{T} = 0.0002\;\text{T} = 2\;\text{gauss}$

For comparison, Earth's magnetic field is about 0.5 gauss — so a typical wire at 1 cm produces a field 4× as strong as Earth's natural field.

Step 2 — At $r = 0.1\;\text{m}$ (10 cm).

The distance grew by 10×, so the field shrinks by 10×:

$B = 0.2 \times 10^{-4}\;\text{T} = 2 \times 10^{-5}\;\text{T} = 0.2\;\text{gauss}$

Step 3 — At $r = 1\;\text{m}$.

Another factor of 10:

$B = 2 \times 10^{-6}\;\text{T} = 0.02\;\text{gauss}$

That's about 1/25th of Earth's field — barely detectable with sensitive instruments.

Step 4 — Determine the direction at a point above the wire.

If the current flows in the $+z$ direction (out of the page in standard physics convention), point your right thumb upward. Your fingers curl counterclockwise when viewed from above.

So at a point directly to the east of the wire, the field points north. To the north of the wire, the field points west. And so on — circulating counterclockwise around the wire.

Step 5 — Force between two parallel wires.

If you put another wire carrying current $I'$ parallel to the first at distance $r$, the second wire feels a force per unit length:

$\dfrac{F}{\ell} = \dfrac{\mu_0\,I\,I'}{2\pi r}$

If both currents flow in the same direction, the force is attractive. If they're antiparallel, repulsive. This is so reliable that the ampere was historically defined by it: 1 A is the current that produces $2 \times 10^{-7}\;\text{N/m}$ of force between two parallel wires 1 m apart.

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Answer: A wire carrying 10 A produces magnetic fields:

• 1 cm away: $B \approx 2 \times 10^{-4}\;\text{T}$ (0.2 mT, ~4× Earth's field)
• 10 cm away: $B \approx 2 \times 10^{-5}\;\text{T}$
• 1 m away: $B \approx 2 \times 10^{-6}\;\text{T}$

The field circulates counterclockwise as viewed when looking along the current direction (right-hand rule), forming concentric circles around the wire. The strength drops as $1/r$.

Try It

• Adjust the current with the slider — the field grows linearly.
• Watch the arrow direction flip when you make the current negative.
• The concentric circles grow stronger near the wire and weaker far away.