Magnetic field of a current: Oersted's experiment and the Biot-Savart law

Activity: Oersted's effect with a matchbox

You need a 1.5 V cell, about 1 m of insulated wire, a small compass and an empty matchbox.

1. 1
   Wind 10 to 15 turns of the wire round the matchbox.
2. 2
   Slide the compass under the turns. Place the box so that the turns run along the needle, north to south.
3. 3
   Touch the wire ends to the cell for a few seconds. Observe: the needle turns away from north-south.
4. 4
   Reverse the cell connections. Observe: the needle turns the other way.
5. 5
   Disconnect the cell. Observe: the needle returns to north-south.
6. 6★
   Result: a current produces a magnetic field around it, and reversing the current reverses the field. Hans Christian Oersted found this in 1820.

★ key step

The Biot-Savart law

Think of a wire as a chain of tiny pieces, each of length Δl. Each piece carrying a current I makes a small field ΔB at a point P a distance r away. Experiments show that ΔB is proportional to the current I, to the length Δl and to sin θ. Here θ is the angle between the piece and the line joining it to P. ΔB is also inversely proportional to r². So ΔB = (μ₀/4π) I Δl sin θ/r². This is the Biot-Savart law. The constant μ₀ is the permeability of free space (vacuum), 4π × 10⁻⁷ T m A⁻¹. Permeability tells us how easily a material lets a magnetic field form in it; air is almost the same as vacuum. In another medium, μ₀ is replaced by that medium's permeability μ. The total field of the wire is the vector sum of the fields of all its pieces. A piece gives no field at points along its own line, where θ = 0.

Derivation: field at the centre of a circular coil

A circular coil of radius r carries a current I. Find the field at its centre O.

1. 1
   Take a small piece Δl of the coil. Its distance from O is r.
2. 2
   The piece is at right angles to the radius, so θ = 90° and sin θ = 1.
3. 3
   Biot-Savart law: ΔB = (μ₀/4π) I Δl/r².
4. 4
   Every piece gives a field at O in the same direction, at right angles to the plane of the coil. So the fields simply add.
5. 5
   The pieces together make the circumference: ΣΔl = 2πr. So B = (μ₀/4π)(I/r²) × 2πr.
6. 6★
   So, B = μ₀I/2r at the centre. For a coil of N turns, B = μ₀NI/2r.

★ key step

Finding the direction: right-hand grip rule and end rule

• Right-hand grip rule: hold the wire in your right hand with the thumb along the current. Your curled fingers show the direction of the field lines.
• Round a long straight wire, the field lines are concentric circles, that is, circles with the wire at their common centre. They lie in a plane at right angles to the wire.
• On paper, a dot ⊙ means a current coming out of the page, and a cross ⊗ means one going into it.
• End rule for a coil: look at one face. If the current there flows clockwise, that face acts as a south pole and B points into it.
• If the current flows anticlockwise, the face is a north pole and B comes out of it. So a current-carrying coil behaves like a short bar magnet.

Worked example: field at the centre of a coil

How to draw the field of a circular coil

• Draw the coil seen edge-on: two small circles one above the other, a distance 2r apart, where the wire cuts the page.
• Put ⊙ in the top circle and ⊗ in the bottom one to show the current.
• Round each small circle, draw closed loops: anticlockwise round ⊙ and clockwise round ⊗.
• Near the centre, draw lines that run almost straight through the coil, along its axis, with arrows.
• Mark the centre O, the radius r, the current I and the field B along the axis at O.
• Label the face that the lines come out of as N, and the other face as S.