Magnetic Field Due to a Current-carrying Conductor: Biot-savart's Law

In one line: A tiny piece of current-carrying wire creates a small magnetic field around it, and the Biot–Savart law tells you exactly how strong that field is and which way it points.

The Real-World Hook

In 1820, Hans Christian Oersted noticed that a compass needle deflected near a current-carrying wire — proving for the first time that electric currents produce magnetic fields. French scientists Jean-Baptiste Biot and Félix Savart followed up that same year with careful experiments to work out the exact mathematical rule connecting current and magnetic field. That rule is the Biot–Savart law — one of the two foundational laws (along with Ampere's law) for calculating magnetic fields from currents

Picture a wire XY carrying current I. Instead of treating the whole wire at once, break it into extremely small pieces called current elements, each of length dl. Pick one such tiny element, and let P be the point in space where you want to know the magnetic field. Let r be the distance from the current element to point P, and let θ be the angle between the direction of current flow (dl) and the line joining the element to P (r).

The magnetic field contribution from just this one tiny element is called dB — a small field, because it comes from a small element.

Experiments show dB depends on four things, combined as follows:

Combining all four gives the scalar (magnitude) form of the Biot–Savart law:

Here, μ₀/4π is simply a constant needed to make the units work out correctly in the SI system.

The direction of dB is perpendicular to the plane containing both dl and r — not along either of them. This is captured using the vector cross product:

Quick rule to find direction: Point your right-hand fingers along the current direction (dl), then curl them toward r; your thumb points in the direction of dB. This is the same right-hand rule used for all cross products in physics, and it's why the field can point "into the page" or "out of the page" depending on geometry.

The Constant μ₀

The proportionality constant has an exact value in SI units:ncert

μ₀ is called the permeability of free space — it plays the same role for magnetic fields that ε₀ (permittivity of free space) plays for electric fields.

When point P lies exactly along the direction of the current element (θ = 0° or 180°), sinθ = 0, so dB = 0. This means no magnetic field is produced directly ahead of or behind a current element — the field only builds up in the surrounding perpendicular directions, reaching its maximum exactly at θ = 90°, where sinθ = 1.

Connecting μ₀, ε₀, and the Speed of Light

There's a striking relationship linking the permeability and permittivity of free space to the speed of light c:

Since c is a fixed universal constant, choosing a value for either μ₀ or ε₀ automatically fixes the other. This connection becomes especially important later when studying electromagnetic waves.

For a current distributed through a volume (rather than a thin wire), the law is rewritten using current density j = I/A:

Dimensions of μ₀: Since μ₀ = N/A², its dimensional formula is [M L T^-2 A^-2].

Given: A current element Δl = Δx î is placed at the origin, carrying current I = 10 A. Find the magnetic field at a point on the y-axis, 0.5 m away, where Δx = 1 cm.

Find: Magnitude and direction of dB.

Solution:
Using ∣dB∣ = \[\frac{\mu_0}{4\pi}\frac{Idl\sin\theta}{r^2}\]​, with dl = Δx = 10^-2 m, I = 10 A, r = 0.5 m, and θ = 90° (since the element is along x and the point is on the y-axis, making sinθ = 1):

Direction: Since dl × r = Δx\[\hat i\] × y\[\hat j\] = y Δx \[\hat k\], the field points in the +z direction (out of the plane).

Answer: dB = 4 × 10^-8 T, directed along +z. Note how small this is — a reminder that individual current elements produce very weak fields; only when integrated over an entire circuit does a measurable field emerge.

• Biot–Savart law: dB = \[\int dB=\frac{\mu_{0}I}{4\pi}\int\frac{dl\sin\theta}{r^{2}}\]​; direction found via right-hand rule on dl × r̂.
• μ₀ = 4π × 10^-7 N A^-2 (permeability of free space); exact fixed value in SI units.
• Field is zero along the current element's own direction (θ = 0° or 180°) and maximum perpendicular to it (θ = 90°).
• Unlike Coulomb's law, the magnetic field depends on the sine of an angle and comes from a vector source, not a scalar one.
• μ₀ε₀ = 1/c² links magnetism, electricity, and the speed of light — a preview of electromagnetic waves.
• For volume currents, use current density j; Idl is replaced by j dV in the integral form.