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Electromagnetic Induction overview

Topic 1 of 6

Flux through a coil

Magnetic flux combines a field with the area it passes through. Flux linkage adds the contributions through a coil's turns. Define the area and its orientation before calculating either quantity.

For a uniform magnetic flux density B and an area A perpendicular to the field, magnetic flux is:

Φ = BA

B describes the local field and is measured in teslas, T. Flux Φ describes the field through the area and is measured in webers, Wb. A field-line drawing helps represent direction and distribution; counting its drawn lines does not determine a numerical flux.

Use the angle to the area normal

A flat surface's normal is a direction perpendicular to its plane. If θ is the angle between B and that normal, the perpendicular projected area is A cos θ:

Φ = BA cos θ

When the plane is perpendicular to B, its normal is parallel to B and flux magnitude is greatest. When the plane is parallel to B, its normal is perpendicular to B and flux is zero. If an acute angle is given to the plane instead, use the complementary angle or the corresponding sine projection.

0° between the field and the area normal

0° between the field and the area normalThe same physical flat area is viewed edge-on. The teal field points right. The selected grey normal makes 0 degrees to the field, on equal horizontal and vertical direction scales. The thick line representing the plane is perpendicular to that normal; the rest of the physical area extends out of this view. With B 0.200 tesla, area 0.00500 square metre and forty equal linked turns, the flux is 0.00100 weber and linkage is 0.0400 weber-turn. Screen line lengths do not measure the area projection.Same B, area A and turn count NBNormal nθ = 0°PlaneΦ = 0.00100 WbΛ = 0.0400 Wb-turn

The plane is perpendicular to B. Its full area is perpendicular to the field, so the flux magnitude is greatest.

60° between the field and the area normal

60° between the field and the area normalThe same physical flat area is viewed edge-on. The teal field points right. The selected grey normal makes 60 degrees to the field, on equal horizontal and vertical direction scales. The thick line representing the plane is perpendicular to that normal; the rest of the physical area extends out of this view. With B 0.200 tesla, area 0.00500 square metre and forty equal linked turns, the flux is 0.000500 weber and linkage is 0.0200 weber-turn. Screen line lengths do not measure the area projection.Same B, area A and turn count NBNormal n60°Plane30°Φ = 0.000500 WbΛ = 0.0200 Wb-turn

The angle to the normal is 60 degrees; the acute angle to the plane is 30 degrees. Use A cos(60 degrees), not A cos(30 degrees).

90° between the field and the area normal

90° between the field and the area normalThe same physical flat area is viewed edge-on. The teal field points right. The selected grey normal makes 90 degrees to the field, on equal horizontal and vertical direction scales. The thick line representing the plane is perpendicular to that normal; the rest of the physical area extends out of this view. With B 0.200 tesla, area 0.00500 square metre and forty equal linked turns, the flux is zero weber and linkage is zero weber-turn. Screen line lengths do not measure the area projection.Same B, area A and turn count NBNormal n90°PlaneΦ = 0 WbΛ = 0 Wb-turn

The plane is parallel to B. Its perpendicular projected area, flux and linkage are zero.

Each panel keeps the same B and physical area while changing the angle between B and the chosen normal. The edge-on plane is perpendicular to that normal; its full area extends out of the view. Use the supplied dimensions and projection, not a screen measurement.

Choosing a positive normal makes flux signed. A field in the opposite direction gives negative flux, and reversing the chosen normal reverses the assigned sign. The physical field does not reverse merely because a reference changes. For a nonuniform field, a single point's B cannot generally represent the whole area without an appropriate approximation.

Sum flux through the turns

For N turns, each linking the same flux Φ, define flux linkage Λ by:

Λ = NΦ = NBA   for perpendicular area
Λ = NBA cos θ   for the stated normal angle

The equal-flux condition matters. If different turns link different fluxes, add their individual fluxes. Increasing N increases linkage for the same imposed field and per-turn area; it does not itself increase that external B.

Worked orientation comparison

Separate one-turn flux from total linkage

A rough field of 0.2 T through about 5 × 10-3 m2 perpendicular to it gives flux of order 10-3 Wb. A few tens of equally linked turns then give a few hundredths of a weber-turn.

Use B = 0.200 T, A = 5.00 × 10-3 m2 per turn and N = 40. At a 60° angle between B and the chosen normal:

Φ = (0.200)(5.00 × 10-3) cos 60°
= 5.00 × 10-4 Wb
Λ = 40Φ = 2.00 × 10-2 Wb-turn

The angle between the plane and B is 30°, not 60°. With the normal instead parallel to B, one-turn flux is 1.00 × 10-3 Wb and linkage is 0.0400 Wb-turn. At a 90° normal angle, both are zero.

Check units and notation

1 Wb = 1 T m2 = 1 V s
= 1 kg m2 s-2 A-1

The turn count N is dimensionless. Writing linkage in Wb-turn makes the counting explicit, but a turn introduces no new physical dimension. Here Λ denotes linkage; it is distinct from Φ for flux through one turn and from symbols used for wavelength or decay elsewhere.

Optional check A 40-turn coil has area 5.00 x 10^-3 m^2 per turn in a uniform 0.200 T field. Every turn links the same flux and its positive normal is at 60 degrees to B. What is the flux linkage?
A 40-turn coil has area 5.00 x 10^-3 m^2 per turn in a uniform 0.200 T field. Every turn links the same flux and its positive normal is at 60 degrees to B. What is the flux linkage?