Magnetic Field due to a Current

Moving charges feel magnetic fields, and they also make them. The Biot–Savart law tells us the field produced by each little piece of a current.

The Biot–Savart law#

A current element i dl⃗ produces at a point P, located by the vector r⃗ drawn from the element to P, a field

where c is the speed of light. The constant 1/ε₀c² is written μ₀, the permeability of vacuum, equal to 4π × 10⁻⁷ T m/A:

dB⃗ is perpendicular to the plane containing the element and P. To find its sense, lay your right palm along dl⃗ with the fingers curling toward r⃗: the thumb points along dl⃗ × r⃗. When the diagram’s plane contains both, the field points into the page (⊗) on one side of the wire and out of it (⊙) on the other.

A straight wire#

Take a straight wire MN carrying current i, and a point P at perpendicular distance d from it (O is the foot of the perpendicular). Every element gives a field in the same direction (into the page, say), so we can just add magnitudes. With l = −d cot θ, dl = d cosec²θ dθ and r = d cosec θ:

On the perpendicular bisector of a segment of length a:

For a very long wire θ₁ → 0 and θ₂ → π:

Right-hand thumb rule: point your right thumb along the current; your curled fingers show the direction of B⃗ round the wire. The field lines of a long straight wire are circles centred on it.

Each current sits in the other’s magnetic field, so two parallel wires push or pull each other. That force is how the ampere is defined.

The force between parallel wires#

Two long parallel wires W₁ and W₂, a distance d apart, carry i₁ and i₂ in the same direction. At an element dl of W₂, the field of W₁ is B = μ₀i₁/2πd (W₂’s own field contributes nothing there). The force on the element is dF = i₂ dl B, and dl⃗ × B⃗ points toward W₁:

The same expression gives the force per length on W₁ due to W₂, as Newton’s third law requires.

Defining the ampere#

For i₁ = i₂ = 1 A and d = 1 m, equation (35.7) gives 2 × 10⁻⁷ N per metre. This was used to define the unit:

Bend a wire into a circle and every element contributes in the same direction at the centre. The result is simple, and far from the loop it becomes a magnetic dipole.

At the centre#

For a loop of radius a, every element is perpendicular to the line to the centre (θ = 90°) and at distance a, and all their fields point the same way. So

Direction: curl your right-hand fingers along the current and your thumb points along B⃗. Equivalently, look along the axis: an anticlockwise current gives a field toward you, a clockwise one a field away from you.

On the axis#

For a point P on the axis at distance d from the centre, each element is at r = √(a² + d²) and perpendicular to r⃗, giving dB = μ₀i dl/4π(a² + d²). Diametrically opposite elements give fields whose sideways parts cancel, leaving only the components along the axis, dB cos α with cos α = a/√(a² + d²). Adding round the loop:

Far away: a magnetic dipole#

When d ≫ a,

and iπa² is the loop’s magnetic moment μ, pointing along the axis with B⃗:

Gauss’s law relates the flux of E through a closed surface to the charge inside. Ampère’s law relates the circulation of B round a closed loop to the current through it.

Circulation and sign conventions#

Draw any closed plane curve and put an arrow on it. Curl the fingers of your right hand along the arrow: the thumb marks the positive side of the enclosed area. (Looking at the loop, an anticlockwise arrow means the positive side faces you.)

At each small step dl⃗ along the curve take B⃗·dl⃗, using the total field there, and add them round the whole loop. This line integral ∮B⃗·dl⃗ is the circulation of B⃗. Currents crossing the area toward the positive side count as positive, the others as negative.

• B⃗ on the left is the field of all currents everywhere.
• i on the right counts only currents passing through the loop; outside currents contribute zero circulation.

A long straight wire, again#

Take a circle of radius d round a long straight wire, in a plane perpendicular to it. By symmetry B⃗ is tangent to the circle and has the same size all round it, so ∮B⃗·dl⃗ = B · 2πd. The current through the circle is i:

Wind wire into a long tight coil and you get a remarkably uniform magnetic field inside and almost none outside: the electromagnet in its simplest form.

The solenoid#

A solenoid is insulated wire wound closely into a helix, usually long compared with its width. Think of it as a stack of circular loops. Two coaxial loops with equal currents give fields that add near the axis but oppose well off it. Stack many and the fields outside cancel more and more while those inside reinforce: a tightly wound long solenoid has a uniform field inside and zero outside.

Apply Ampère’s law to a rectangle PQRS with side PQ (length l) inside, parallel to the axis, and the opposite side outside. B⃗·dl⃗ is Bl along PQ and zero on the other three sides (B is zero outside, and perpendicular to the two short sides inside). If there are n turns per unit length, nl turns, each carrying i, pass through the rectangle:

Strictly this is for an infinitely long solenoid, but it holds well in the middle part of any solenoid more than five or six times as long as its diameter.

The toroid#

Bend a solenoid into a ring and join its ends (or wind wire closely round a ring) to get a toroid. Take a circle of radius r inside the winding, concentric with the ring. By symmetry B is tangent to it and the same all round, so ∮B⃗·dl⃗ = 2πrB. With N turns in all, the current through the circle is Ni: