EM-13 · Magnetostatics field

The Biot–Savart Law

A magnetic field is made by moving charge — by current. The Biot–Savart law gives the tiny field dB that a current element makes at a point. Rotate the test point to see dB stay perpendicular to both the current and the line to it, growing as sinθ.

Rotate the test point to see dB

Rotate the test point around a current element flowing upward. How big is the tiny field dB along the current (above and below)? And to the side, perpendicular to it? Which way does dB point?

Test point angle θ (from the current)θ = 50°

Drag to orbit. The slider sets the test point's angle θ.

The tiny field dB at this point

dB = (μ₀/4π)(I dl sinθ)/r² · perpendicular to both current and distance

B = ∫ dB = (μ₀/4π)∫ I dl × r̂ / r² · integrate along the current path

dB magnitude (vs maximum) · 77%

Slanted

A magnetic field comes from current

A charge at rest makes an electric field (EM-03). When charge moves — when a current flows — a new field appears around it: the magnetic field B. Hold a compass needle beside a wire and it swings only while current flows. The magnetic field comes not from the amount of charge but from its motion. Even a magnet, in fact, is made by the flow of electrons inside its atoms.

Biot–Savart, the tiny field

A short piece cut from the current's path is a current element I dl. The Biot–Savart law gives the tiny field this one piece makes at a point a distance r away: dB = (μ₀/4π) I dl × r̂ / r². Its magnitude falls as the inverse square of distance and scales with sinθ (θ being the angle between the current and the line to the point). Its direction is perpendicular to both the current and that line, set by the right-hand rule. The constant μ₀ is the permeability of free space.

ObservedB ∝ I dl sinθr²

dB scales with sinθ and inversely with the square of distance.

ChoosedB ∝ dl × ?

The direction of dB is the cross product of dl and r̂.

Fill indB = (μ₀4π) ?

The Biot–Savart law gives one piece's dB.

The field circles the current

The electric field radiated straight out from charge. The magnetic field is different. Because of the cross product dl × r̂, dB wraps around the current as an axis. Along the current, above and below (θ=0), sinθ=0 so dB vanishes; to the side, perpendicular to it (θ=90°), it is strongest. So the field around a straight current forms concentric circles. Point your right thumb along the current and your curling fingers show the way the field circulates.

The whole field = an integral

Once you know one piece's dB, you add the dB of every piece as vectors along the whole current path to get the total field: B = ∫ dB. It is the same structure as integrating a point charge's dE over a charge distribution in EM-04. For an infinitely long straight current, for instance, this integral comes out cleanly as B = μ₀I/(2πr), so the field falls inversely with distance. When symmetry is good, the next unit's Ampère law lets you skip even this integral.

On your ownB = ?

The whole field integrates dB along the current.

Back to the first screen

On the first screen, moving the test point along the current, above and below, made dB vanish to zero; moving it to the side, perpendicular, made it largest. Its size followed sinθ. And dB always pointed out of the plane formed by the current and the test direction, wrapping around the current. That is the cross product dl × r̂ at work. Adding this one piece’s dB all along the current path gives the whole field B.

The Biot–Savart law gives the tiny field dB that a current element I dl makes at a point: dB = (μ₀/4π) I dl × r̂ / r², perpendicular to both the current and the line to the point (right hand), inversely proportional to the square of distance, and proportional to sinθ. It is zero along the current and largest perpendicular to it. The total field is the integral along the current path, B = ∫ dB.

What comes next

The Biot–Savart integral works for any shape of current but takes effort. When symmetry is good there is a shortcut: Ampère's law (EM-14), that the field summed around a closed loop, ∮B·dl, is proportional to the current threading the loop. Just as Gauss's law did for electrostatics, it pulls out the field of a wire, solenoid, or toroid without integrating.