Moving charges create magnetic fields; Biot–Savart gives the field of each small current element, and Ampere's law gives the field directly when the current distribution is symmetric.
How it's asked: Field at the centre and on the axis of a circular loop, field of a long straight wire, field inside a solenoid and toroid, and direction by the right-hand rule.
Why this is true
A small current element I dl produces a field proportional to I dl sinθ/r², perpendicular to both dl and the line to the point — so there is no field along the wire's own line. Adding these contributions around a circular loop, every element is at the same distance from the centre and contributes in the same direction, giving μ₀I/2R. For long straight wires and solenoids, Ampere's law is easier: the line integral of B around a closed loop equals μ₀ times the current threading it. Around a long wire B is the same on a circle, giving μ₀I/2πr; inside a long solenoid the field is uniform and nearly zero outside, so B = μ₀nI. A toroid is a solenoid bent into a ring, so its field is confined inside the core.
Key points
Key formulae
Biot–Savart
dB = (μ₀/4π) I dl sinθ/r²
Straight wire
B = μ₀I/(2πr)
Circular loop (centre)
B = μ₀NI/(2R)
Ampere's law
∮B·dl = μ₀I(enclosed)
Solenoid
B = μ₀nI
Toroid
B = μ₀NI/(2πr)
Magnetic fields of standard current distributions
| Source | Field | Depends on |
|---|---|---|
| Long straight wire | μ₀I/2πr | 1/r |
| Circular loop, centre | μ₀NI/2R | 1/R |
| Circular loop, far on axis | μ₀NIR²/2x³ | 1/x³ (dipole-like) |
| Long solenoid, inside | μ₀nI | Turns per length only |
| Toroid, inside core | μ₀NI/2πr | Mean radius |
Common traps
- In B = μ₀nI, n is turns per metre, not the total number of turns.
- A point on the line of a straight current element gets no field from it (sinθ = 0).
Worked example
A solenoid 0.5 m long has 500 turns and carries 2 A. Find the magnetic field inside it.
High-yield
Wire μ₀I/2πr; loop centre μ₀NI/2R; solenoid μ₀nI; toroid μ₀NI/2πr.
Quick check
Q1.The magnetic field 5 cm from a long straight wire carrying 10 A is:
Q2.The magnetic field inside a long ideal solenoid is: