Physics › Magnetic fields › Magnetic flux and flux linkage
Magnetic flux and flux linkage
Before induction can be stated, it needs its own quantity. Flux counts the field threading an area; flux linkage multiplies by the turns of a coil; and one cosine, of the angle to the coil's normal, handles every tilt in between.
Pick your board and the few notes written for the other boards quietly fold away, here and in the practice players. Nothing is deleted: every folded piece reopens on a tap.
Builds on Magnetic flux density and the force on a wire.
IN THIS TOPIC
- Use Φ = BA for an area normal to the field, in webers.
- Use flux linkage NΦ, and NΦ = BANcosθ for a rotated coil, with θ to the normal.
- Describe the required-practical investigation with a search coil and oscilloscope.
COMMON MISCONCEPTION
θ is the angle between the field and the coil.
Flux: field through an area
Picture the field as flow lines and ask how much passes through a chosen area. The magnetic flux through an area A held face-on to a field B, with B along the area's normal, is
measured in webers, where one weber is one tesla times one square metre. The name explains the other unit. Flux density B is flux per unit area, webers per square metre, and that is exactly what a tesla is.
WORKED EXAMPLE
Flux, then linkage
A 200-turn coil of area 3.0 × 10−3 m2 sits face-on to a 0.15 T field. Find the flux through one turn and the flux linkage.
Face-on means the normal lies along the field, so the cosine is 1 and Φ = BA = 0.15 × 3.0 × 10−3 = 4.5 × 10−4 Wb.
Linkage counts every turn, so NΦ = 200 × 4.5 × 10−4 = 0.090 Wb turns.
The two quantities differ only by the turn count, but the induction unit ahead runs on the linkage, because each turn contributes its own share of emf.
Linkage, and the tilt
A coil of N turns threads the same flux N times over, and the quantity driving every result in electromagnetic induction is the flux linkage, NΦ. Tilt the coil and only the component of B along the coil's normal still counts.
where θ is the angle between B and the normal to the coil's plane, and that angle, misread, is the most reliable source of error in this topic. Face-on means θ = 0 and full linkage; edge-on means θ = 90° and none, even though the coil's plane then lies along the field. Check any answer against those two limits before trusting it.
GUIDED PRACTICE
Tilt it thirty degrees
The same coil is tilted so its normal makes 30° with the field. Find the new flux linkage before opening the working, and check it against the limits.
Show the working
NΦ = BANcos θ = 0.090 × cos 30° = 0.078 Wb turns.
The limits bracket it. Face-on gives 0.090 and edge-on gives zero, so a thirty-degree tilt ought to cost little, and cos 30° = 0.87 delivers exactly that. An answer built on sin would have thrown most of the linkage away for a thirty-degree nudge, failing the sanity check on sight.
Required practical 11, the search coil
The eleventh required practical measures how linkage depends on orientation, using a search coil, a small many-turned coil on a handle, connected to an oscilloscope. Place it in an alternating field and the changing linkage induces an alternating pd whose amplitude on the screen is proportional to the flux linkage through the coil. Rotate the search coil in steps, read the amplitude at each angle, and a plot of amplitude against cosθ comes out straight. That is the cosine law confirmed by experiment, and the induction law it leans on arrives properly in the next lesson.
ASSESSMENT FOCUS
- Φ = BA needs B along the normal. For anything else go straight to BANcosθ, and say what θ is measured to.
- The limits are your self-check. Face-on, θ = 0, gives maximum linkage; edge-on, θ = 90° with the field lying in the coil's plane, gives zero. Quote them when you explain.
- Units earn marks here. Flux in webers, flux linkage in weber-turns, and one tesla is one weber per square metre.
- Read “rotated by” carefully. A coil rotated 30° from face-on has θ = 30°, while one rotated 30° from edge-on has θ = 60°. Draw the normal before you substitute anything.
- In the practical the oscilloscope's amplitude stands in for the linkage, and plotting it against cosθ to get a straight line through the origin is the analysis the mark scheme wants.
CHECK YOURSELF
A 500-turn coil of area 2.0 × 10−3 m² sits in a 0.15 T field with its normal at 60° to the field. Find the flux linkage, and state what it becomes face-on.
Show a hint
BANcosθ, with θ to the normal; then set θ to zero.
Show the answer
= 0.15 × 2.0 × 10−3 × 500 × cos 60° = 0.075 Wb turns.
Face-on, cosθ = 1, so the linkage doubles to 0.15 Wb turns.
Edge-on it would fall to zero. Those two limits catch any misread angle.
Flux is field through area, linkage multiplies by the turns, and the cosine's angle always runs to the normal.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their mark schemes on the magnetic flux and flux linkage questions page.
WHERE TO GO NEXT
- Required practical 11: flux linkage and a search coil puts this topic in the lab, and the written papers ask about it.
- Trigonometry and resolving vectors is the maths this lesson leans on, worked through from GCSE.
CHECK YOUR PROGRESS
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- Use Φ = BA for an area normal to the field, in webers.
- Use flux linkage NΦ, and NΦ = BANcosθ for a rotated coil, with θ to the normal.
- Describe the required-practical investigation with a search coil and oscilloscope.
Open the full revision checklist to track your progress across the whole unit.