I is the current, R the loop's radius and x the distance from the centre along the axis.
Derivation
- Each element is √(R² + x²) away and perpendicular to that line, so dB = (μ₀/4π) I dl/(R² + x²).
- The perpendicular components cancel; the axial components dB × R/√(R² + x²) add.
- B = (μ₀/4π) I (2πR) R/(R² + x²)^(3/2) = μ₀IR²/2(R² + x²)^(3/2).
Points to remember
- At x = 0 it becomes μ₀I/2R.
- For x ≫ R: B ≈ μ₀IR²/2x³ = (μ₀/4π)(2m/x³), with m = IπR², like a bar magnet.
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 circular loop on its axis?
B = μ₀IR²/(2(R² + x²)^(3/2)). I is the current, R the loop's radius and x the distance from the centre along the axis.
How do you derive the magnetic field due to a circular loop on its axis?
Each element is √(R² + x²) away and perpendicular to that line, so dB = (μ₀/4π) I dl/(R² + x²). The perpendicular components cancel; the axial components dB × R/√(R² + x²) add. B = (μ₀/4π) I (2πR) R/(R² + x²)^(3/2) = μ₀IR²/2(R² + x²)^(3/2).