Electromagnetism: (b) magnetic flux linkage

The idea

Magnetic flux linkage extends magnetic flux Φ to a coil of N turns: each turn links the same flux, so linkage NΦ is measured in webers (Wb) or weber-turns. Flux itself is Φ = BA cos θ, where B is flux density in tesla, A is area in m² and θ is the angle between B and the normal to the area. If B = 0.20 T, A = 3.0 × 10⁻³ m² and the field is perpendicular to the coil, Φ = 6.0 × 10⁻⁴ Wb; a 250-turn coil then has linkage NΦ = 0.15 Wb. Linkage changes when B, A or orientation changes, and this changing linkage drives induced e.m.f. in the next specification point.

What earns the marks

• Magnetic flux linkage is defined as NΦ, the product of the number of turns N and the magnetic flux Φ through one turn.
• Magnetic flux Φ = BA cos θ, with B in tesla, A in m² and θ measured between B and the normal to the area.
• The unit of flux and of flux linkage is the weber (Wb); flux linkage may also be quoted in weber-turns.
• Flux linkage changes if B changes, if the area A changes, or if the angle θ between field and normal changes.
• A numerical check: B = 0.20 T, A = 3.0 × 10⁻³ m², N = 250 gives Φ = 6.0 × 10⁻⁴ Wb and NΦ = 0.15 Wb.

Common mistakes and how to put them right

• Treating flux linkage as simply BA and forgetting to multiply by the number of turns N.

  Put it right: Use NΦ = NBA cos θ.

• Measuring θ from the plane of the coil instead of from the normal to the area.

  Put it right: θ is the angle between B and the normal, so a field parallel to the plane gives zero flux.

• Using area in cm² without converting to m².

  Put it right: Convert, for example 30 cm² = 3.0 × 10⁻³ m², before multiplying.

Exam tips

• Write the defining equation NΦ = NBA cos θ before substituting values so the role of each symbol is clear.
• Check that area is in m² and that the angle used is measured from the normal to the coil.
• When a question asks for a change in flux linkage, calculate the initial and final values separately and subtract, rather than assuming the change equals the final value.