Short on time? This page is the recap. For the diagrams, the derivations and the worked examples, use the full study guide. The surface-integral form is part of the calculus-based course.
Recap
- Magnetic flux measures how much magnetic field passes through a surface. It is a scalar, measured in webers: 1 Wb = 1 T·m².
- The area vector A is perpendicular to the surface, with size equal to the area. For an open surface you choose its direction; on a closed surface it points outward.
- The sign of the flux comes from the dot product: positive when B has a component along A, negative when B points against A, zero when B runs along the surface.
- When B varies over the surface, split the surface into strips or rings where B is constant and integrate.
- μ₀ = 4π × 10⁻⁷ T·m/A.
Key relationships
| Situation | Flux |
|---|---|
| Uniform B, flat surface | Φ_B = B·A = BA cos θ |
| Surface in the xy-plane, A = A k̂ | Φ_B = B_z A |
| General surface or varying B | Φ_B = ∫B·dA |
| Rectangle beside a long wire (near side d, width b, length ℓ) | Φ_B = (μ₀Iℓ/2π) ln[(d + b)/d] |
| B depends only on radius r, flat disc | Φ_B = ∫B(r) 2πr dr |
| One turn of a long solenoid, radius R | Φ_B = μ₀nI πR² |
| Any closed surface | ∮B·dA = 0 |
Assumptions
- “Uniform” means B has the same size and direction at every point of the surface.
- The ideal long solenoid has uniform B inside and B ≈ 0 outside.
- “Long” wires are much longer than the distances involved.
Mistakes to avoid
- Angle from the surface. If B makes 25° with the plane, use θ = 65°.
- Dropping the sign. A half turn changes +BA into −BA; the change is 2BA.
- Too much area. Around a solenoid, only the solenoid’s own cross-section carries flux.
- Middle-value shortcut. For B ∝ 1/x, the field at the middle times the area is wrong; integrate.
- Zero net flux ⇒ zero field. Wrong: it means every line that enters also leaves.
Quick self-check
- A flat surface of area 0.050 m² is perpendicular to a uniform 0.20 T field. What is the flux through it? (1.0 × 10⁻² Wb)
- The same surface is turned so that its plane lies along the field. What is the flux now? (Zero: θ = 90°)
- What is the net magnetic flux through a closed box that surrounds the north end of a bar magnet? (Zero, as for every closed surface)
Next: practice questions.