n is the number of turns per unit length (N/L) and I the current.
Derivation
- Ampère's law round a rectangle with one side of length l inside the solenoid and one outside.
- The field outside is nearly zero and the short sides are perpendicular to B, so only the inside side counts: Bl.
- The rectangle encloses nl turns, so Bl = μ₀nlI and B = μ₀nI.
Points to remember
- The field inside is uniform, and at each end it is μ₀nI/2.
- With a core of relative permeability μᵣ: B = μ₀μᵣnI.
All field formulas
| Field due to | Formula |
|---|---|
| Electric field due to a point charge | E = kq/r² = q/(4πε₀r²) |
| Electric field due to an electric dipole on its axis | E = 2kp/r³ |
| Electric field due to an electric dipole on its equatorial line | E = kp/r³ |
| Electric field due to a charged ring on its axis | E = kQx/(x² + R²)^(3/2) |
| Electric field due to an infinite line charge | E = λ/(2πε₀r) = 2kλ/r |
| Electric field due to an infinite charged sheet | E = σ/(2ε₀) |
| Electric field due to a charged conducting sphere (or spherical shell) | E = kQ/r² outside, 0 inside |
| Electric field due to a uniformly charged non-conducting sphere | E = kQr/R³ inside, kQ/r² outside |
| Magnetic field due to a long straight wire | B = μ₀I/(2πr) |
| Magnetic field due to a finite straight wire | B = (μ₀I/4πd)(sin α + sin β) |
| Magnetic field due to a circular loop at its centre | B = μ₀I/(2R) |
| Magnetic field due to a circular loop on its axis | B = μ₀IR²/(2(R² + x²)^(3/2)) |
| Magnetic field due to a long solenoid | B = μ₀nI |
| Magnetic field due to a toroid | B = μ₀NI/(2πr) |
Questions students ask
What is the magnetic field due to a long solenoid?
B = μ₀nI. n is the number of turns per unit length (N/L) and I the current.
How do you derive the magnetic field due to a long solenoid?
Ampère's law round a rectangle with one side of length l inside the solenoid and one outside. The field outside is nearly zero and the short sides are perpendicular to B, so only the inside side counts: Bl. The rectangle encloses nl turns, so Bl = μ₀nlI and B = μ₀nI.