5.1 Sources of Magnetic Fields

Biot-Savart Law

The Biot-Savart Law gives the magnetic field created by a small current element: dB = (μ₀/4π)(I dl × r̂)/r². The field is a vector cross product, so it's perpendicular to both the current direction and the line to the field point. Integrate over the entire current path to find the total field. This is the magnetic analog of Coulomb's law for point charges.

Field from a Long Straight Wire

Integrating the Biot-Savart Law for an infinite straight wire gives B = μ₀I/(2πr), where r is the perpendicular distance from the wire. The field forms concentric circles around the wire; direction is found with the right-hand rule (thumb points along current, fingers curl in field direction).

Field at the Center of a Loop

For a circular loop of radius R carrying current I, the field at the center is B = μ₀I/(2R), directed along the axis per the right-hand rule. Superposition applies: fields from multiple wires or loop segments add as vectors. This principle extends to solenoids and more complex current distributions.

Key terms

Biot-Savart Law

dB = (μ₀/4π)(I dl × r̂)/r² — gives the magnetic field contribution from a small current element.

Magnetic field of a long straight wire

B = μ₀I/(2πr), forming concentric circles around the wire; direction by right-hand rule.

Magnetic field at center of circular loop

B = μ₀I/(2R), directed along the loop's axis.

Right-hand rule (for B field from current)

Point thumb in current direction; curled fingers show the direction of the magnetic field lines.

Superposition principle (magnetic fields)

The net magnetic field from multiple current sources is the vector sum of each individual field.

Try an AP-style question

What law gives the magnetic field produced by a small current element?

1. Lenz's law
2. The Biot–Savart law
3. Coulomb's law
4. Gauss's law