Library / Foundations: Physics / FPH-59
Foundations: Physics
Magnetic flux through a flat surface
Magnetic flux through a flat area A in a uniform field B, where θ is the angle between the field and the normal (the line at right angles) to the surface.
- Ref
- FPH-59
- Rev
- v1
- Field
- Foundations: Physics
- Result
- Φ [Wb]
B is the flux density in tesla, A the area in m², and θ the angle between the field direction and the normal to the surface (not the surface itself). At θ = 0° the field goes straight through and the flux is BA; at θ = 90° the field skims along the surface and the flux is zero. For a coil with N turns the flux linkage is NΦ.
Magnetic flux through a flat area A in a uniform field B, where θ is the angle between the field and the normal (the line at right angles) to the surface. It is largest when the field passes straight through the surface and zero when it runs along it.
Calculator
- Flux when face-on, BA0.01 Wb
- Flux linkage of a 100-turn coil, 100Φ0.866 Wb
- cos θ0.866
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 30° from the normal the surface catches 86.6 % of the maximum flux, 8.66 mWb.
- A generator coil turning in a field changes θ steadily, so Φ = BA cos ωt, and Faraday’s law (the induced emf page) gives the alternating voltage.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Wb) | Local step | Change in result |
|---|---|---|---|---|---|---|
| A Area of the surface | 1e-4 to 5 m² | 1.732e-5 | 0.866 | 0.866 | +10% | +10 % |
| B Magnetic flux density | 0.001 to 5 T | 4.33e-5 | 0.2165 | 0.2165 | +10% | +10 % |
| θ Angle between the field and the surface normal | 0 to 180 ° | −0.01 | 0.01 | 0.02 | +10% | −3.16 % |
Sweep of Angle between the field and the surface normal
| Angle between the field and the surface normal (°) | Magnetic flux (Wb) |
|---|---|
| 0 | 0.01 |
| 22.5 | 0.009239 |
| 45 | 0.007071 |
| 67.5 | 0.003827 |
| 90 | 6.123e-19 |
| 112.5 | −0.003827 |
| 135 | −0.007071 |
| 157.5 | −0.009239 |
| 180 | −0.01 |
Sweep one variable
Variables and how each one contributes
Magnetic flux
What the equation gives you. Everything else feeds it.
Magnetic flux density
The magnetic flux density, in tesla. Raises the flux in direct proportion. Raising it by 10 % (to 0.22) moves the result by +10 %, from 0.00866 to 0.009526 Wb.
Area of the surface
The area of the flat surface, in m². Raises the flux in direct proportion. Raising it by 10 % (to 0.055) moves the result by +10 %, from 0.00866 to 0.009526 Wb.
Angle between the field and the surface normal
The angle between the field and the surface normal, in degrees. Lowers the flux as the surface tips away from face-on, because cos θ falls; the effect grows towards 90°. Raising it by 10 % (to 33) moves the result by −3.16 %, from 0.00866 to 0.008387 Wb.
Worked example with the values above
- Angle factor: cos 30° = 0.866025.
- Face-on flux: BA = 0.2 × 0.05 = 0.01 Wb.
- Φ = 0.01 × 0.866025 = 0.00866025 Wb.
Assumptions and limits
- A uniform field over a flat surface.
- θ measured from the surface normal.
Common mistakes
- Measuring the angle from the surface instead of from its normal (which swaps sine and cosine).
- Using a calculator in radian mode with a degree angle.
- Forgetting to multiply by the number of turns when finding flux linkage.
Statistics and uncertainty
The flux depends on cos θ, so near θ = 90° it is very sensitive to alignment: at 85° a 1° error changes the flux by about 20 %, while near 0° the same error changes it by under 0.02 %.
Where it is used
- Setting up Faraday’s law for coils, generators and transformers.
- Interpreting search-coil and fluxgate sensor readings.
Maths you need
- Magnetic field: flux density in tesla
- Cosine: projection of an area