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Magnetic flux through a coil

How much magnetic flux links a coil sitting in a field of a given strength, area and angle?

The formula
Φ=B⁢A⁢cosθ
Φ
Magnetic flux, Wb
B
Field strength, T
A
Area, m²
θ
Angle from the normal, °
LaTeX 
Φ = B \cdot A \cdot \cos(θ)

Work it out

G

The magnetic flux density at the coil. Core datasheets of the period are quoted in gauss.

cm²

The cross-sectional area the field passes through.

°

The angle between the field and the coil's normal. Zero is face-on, where linkage is greatest.

Compare with a variation

Result

Φ Magnetic flux 500 µWb

The flux linking the coil. Its rate of change, not its size, is what induces a voltage.

With your numbers
500 µWb=0.5 T⁢10 cm²⁢cos0 °
LaTeX 
500\,\mathrm{µWb} = 0.5\,\mathrm{T} \cdot 10\,\mathrm{cm²} \cdot \cos(0\,\mathrm{°})

This result is a link — the address bar holds your numbers, so it can be pasted into a post and opened to the same answer.

What this looks like

Magnetic flux against field strength Field strength swept from 2500 G to 7500 G with everything else held at your numbers. The dashed lines cross where you are.
Magnetic flux against field strengthMagnetic flux rises from 250 µWb to 750 µWb as field strength rises from 2500 G to 7500 G. At your field strength of 5000 G it is 500 µWb.20040060080020004000600080005000 G500 µWbField strength (G)Magnetic flux (µWb)
The formula behind the curve
Φ=B⁢A⁢cosθ
Φ
Magnetic flux, Wb
B
Field strength, T
A
Area, m²
θ
Angle from the normal, °
LaTeX 
Φ = B \cdot A \cdot \cos(θ)
What moves the answer Each input moved 10% either way, with the others held still, and the effect on magnetic flux.
What moves the answerMagnetic flux is most sensitive to Field strength, which moves it by about 10% for a 10% change. It is least sensitive to Angle, at about 0.0123%.Change in the answer when each input moves by 10%-20%-10%10%20%Field strength±10Coil area±10Angle±0.0123
The formula behind the curve
Φ=B⁢A⁢cosθ
Φ
Magnetic flux, Wb
B
Field strength, T
A
Area, m²
θ
Angle from the normal, °
LaTeX 
Φ = B \cdot A \cdot \cos(θ)

Method

  1. Convert the field to tesla and the area to square metres. Flux is a product of the two in SI, and gauss and square centimetres are both a hundred-to-one from that.
  2. Convert the angle from degrees to radians.
  3. Multiply the field by the area and by the cosine of the angle. Face-on (0°) gives the full product; edge-on (90°) gives zero, however strong the field is.

Assumptions

  • The field is uniform across the coil's area. A real EPG field is not, especially near a pole face; this treats the coil as small enough, or the field as even enough, that one number describes it.
  • The coil is a flat loop (or a stack of identical turns, each contributing the same flux) rather than a wound solenoid threading its own field.
  • No fringing or leakage is modelled. A real magnetic circuit loses some flux around the edges of the gap; this is the ideal figure.