Magnetic Effects of Current

Syllabus Unit: Electrodynamics

Why it matters

Chapter overview

Moving charges and currents generate magnetic fields, and magnetic fields in turn exert forces on moving charges and currents — the Biot-Savart law and Ampere's circuital law give two complementary routes to calculate these fields for wires, loops, solenoids and toroids, while the force laws F = qv×B and F = IL×B explain motors, cyclotrons and every current-carrying conductor placed in a field.

Physics Syllabus Electrodynamics 25%–30% weightage

Live Animation1 / 4

1Current ElementA small segment I·dl carries current

2Biot-Savart LawdB = (μ0/4π) I dl sinθ / r²

3Integrate Along the WireSum dB contributions from every element

4Straight Wire ResultB = μ0I / 2πa

1Biot-Savart Law & the Straight Wire

The Biot-Savart law gives the tiny magnetic field contributed by an infinitesimal current element — integrating it over a straight wire yields the field around any current-carrying conductor.

Learning Objectives

State the Biot-Savart law dB = (μ0/4π) I dl × r̂ / r² and identify each term.
Apply the right-hand thumb rule to find the direction of B around a straight wire.
State the field of an infinitely long straight wire, B = μ0I/2πa.
Compare the field of a finite wire (involving sinθ1 + sinθ2) with the infinite-wire limit.

Key Terms

Biot-Savart LawdB = (μ0/4π)(I dl × r̂)/r², giving the magnetic field due to a small current element; μ0 = 4π×10^-7 T·m/A is the permeability of free space.

Right-Hand Thumb RulePoint the thumb along the conventional current; the curled fingers give the direction of the circular magnetic field lines around the wire.

Permeability of Free Space (μ0)The constant relating current to the magnetic field it produces in vacuum, μ0 = 4π×10^-7 T·m/A.

Perpendicular Distance (a)The shortest distance from the field point to the wire, appearing in B = μ0I/2πa for an infinite straight wire.

The **Biot-Savart law** is the magnetic analogue of Coulomb's law: instead of the field due to a point charge, it gives the tiny field dB due to an infinitesimal current element I·dl.

dB = (μ0/4π) · I dl sinθ / r²

μ0/4π = 10^-7 T·m/A exactly.
dl = the current element vector, in the direction of current flow.
r = distance from the element to the field point; θ = angle between dl and r.
Direction is given by dl × r̂ — into or out of the page, found using the right-hand rule.

Field of a straight wire

B = μ0 I / 2πa (infinite straight wire, perpendicular distance a)

Key PointFor a finite wire, B = (μ0I/4πa)(sinθ1 + sinθ2), where θ1 and θ2 are the angles subtended by the two ends at the field point. This reduces to μ0I/2πa as the wire becomes very long (θ1, θ2 → 90°).

Common MistakeTrap: the magnetic field lines around a straight wire are concentric circles, not radial like an electric field's — and B falls off as 1/a, not 1/a² like Coulomb's law.